%!PS-Adobe-2.0 %%Creator: dvips(k) 5.95b Copyright 2005 Radical Eye Software %%Title: G:/F_USA_neu/Data1/UNI/INSTITUT/VORLESUN/WS01_02/advalg.dvi %%CreationDate: Sat Jun 14 12:33:42 2008 %%Pages: 104 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMR17 CMR12 CMCSC10 CMMI12 CMR9 CMBX12 CMTI12 CMMI8 %%+ CMSY10 CMSY8 MSAM10 MSBM10 MSBM7 CMEX10 CMR8 LINE10 CMMI6 CMSY6 CMR6 %%+ CMMI10 EUFM10 EUFM7 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: "C:\Program Files\MiKTeX 2.5\miktex\bin\dvips.exe" %+ G:/F_USA_neu/Data1/UNI/INSTITUT/VORLESUN/WS01_02/advalg.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2008.06.14:1233 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{ dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def} if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def} def end %%EndProcSet %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 84 /T put dup 103 /g put dup 104 /h put dup 109 /m put dup 112 /p put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put dup 110 /n put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 8 /braceleftbig put dup 9 /bracerightbig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 40 /braceleftBigg put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 81 /producttext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 101 /tildewide put dup 102 /tildewider put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 126 /tilde put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 10 /circlemultiply put dup 48 /prime put dup 67 /C put dup 68 /D put dup 70 /F put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 77 /M put dup 78 /N put dup 82 /R put dup 88 /X put dup 97 /a put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 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LINE10 1.001 %%CreationDate: 1992 Oct 23 20:22:05 %%RevisionDate: 2001 Jun 05 20:22:05 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LINE10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /LINE10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /a0 put dup 1 /a1 put dup 8 /a8 put dup 9 /a9 put dup 11 /a11 put dup 17 /a17 put dup 21 /a21 put dup 25 /a25 put dup 27 /a27 put dup 42 /a42 put dup 43 /a43 put dup 45 /a45 put dup 51 /a51 put dup 54 /a54 put dup 63 /a63 put dup 64 /a64 put dup 65 /a65 put dup 72 /a72 put dup 80 /a80 put dup 81 /a81 put dup 82 /a82 put dup 85 /a85 put dup 88 /a88 put dup 106 /a106 put dup 113 /a113 put dup 115 /a115 put dup 122 /a122 put readonly def /FontBBox{-150 -150 1020 1020}readonly def currentdict end currentfile eexec D9D66F637A9E5292A4933615152D29EEC26E1BED2E48CAB7AC058698EA30B07E 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 8 /Phi put dup 9 /Psi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 65 /A put dup 72 /H put dup 75 /K put dup 77 /M put dup 78 /N put dup 79 /O put dup 83 /S put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 126 /tilde put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /squaremultiply put dup 3 /square put dup 106 /subsetdblequal put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 10 /circlemultiply put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 75 /K put dup 102 /braceleft put dup 103 /braceright put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 32 /arrowleft put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 75 /K put dup 77 /M put dup 80 /P put dup 81 /Q put dup 83 /S put dup 86 /V put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put dup 113 /coproduct put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 19 /iota put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 44 /arrowhookleft put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 122 /z put readonly def /FontBBox{14 -250 1077 750}readonly def currentdict 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 3 /Lambda put dup 8 /Phi put dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 22 /macron put dup 25 /germandbls put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 40258431 52099146 1000 600 600 (G:/F_USA_neu/Data1/UNI/INSTITUT/VORLESUN/WS01_02/advalg.dvi) @start /Fa 146[48 109[{}1 49.8132 /EUFM7 rf /Fb 143[35 2[53 109[{}2 66.4176 /EUFM7 rf /Fc 180[39 75[{}1 49.8132 /MSBM7 rf /Fd 143[50 2[76 4[52 50 18[67 84[{}5 99.6264 /EUFM10 rf /Fe 145[50 4[29 105[{}2 83.022 /CMMI10 rf /Ff 236[61 61 18[{}2 83.022 /CMEX10 rf /Fg 129[30 74[30 30 30 30 6[24 24 40[{}7 49.8132 /CMR6 rf /Fh 185[44 1[48 34 18[18 37[48 6[32 2[48{}7 49.8132 /CMSY6 rf /Fi 140[29 29 1[32 31 38 2[33 25 22 2[30 4[34 8[50 5[46 3[48 58 8[50 44 46 46 5[19 59[{}19 49.8132 /CMMI6 rf /Fj 133[83 6[83 1[83 6[83 17[83 2[42 2[83 83 83 7[83 6[42 83 83 8[83 2[83 5[83 1[83 83 14[83 1[83 3[42 3[83 5[42 1[83 83 6[42 83{}27 83.022 /LINE10 rf /Fk 129[35 7[37 39 27 28 28 2[35 39 59 20 37 1[20 1[35 22 31 39 31 39 35 3[20 1[20 7[39 3[55 53 65 1[55 2[53 6[53 3[55 8[35 35 35 35 35 4[55 1[27 27 30[55 51 6[59 1[{}40 66.4176 /CMR8 rf /Fl 153[100 55 3[127 94 6[127 144 4[83 1[94 105 2[151 111 35[80 13[75 6[73 73 8[58 58 8[{}17 99.6264 /CMEX10 rf /Fm 165[44 7[48 52 2[48 2[52 7[48 67[{}6 66.4176 /MSBM7 rf /Fn 165[66 7[72 77 2[72 2[77 7[72 30[77 27[77 8[{}8 99.6264 /MSBM10 rf /Fo 149[77 102[77 77 2[{}3 99.6264 /MSAM10 rf /Fp 141[59 3[35 6[35 35 26[54 3[42 50 1[55 37 12[0 3[47 71 19 23[55 2[55 55 9[55 6[35 55 1[55{}20 66.4176 /CMSY8 rf /Fq 141[83 75 2[50 3[28 39 39 50 50 6[66 66 1[66 66 4[61 2[60 1[81 69 2[120 1[76 2[84 59 72 53 77 52 7[50 1[55 55 0 0 2[66 66 100 7[100 100 6[100 100 5[77 1[77 2[77 77 77 77 3[50 3[77 1[77 1[77 2[50 77 28 77{}50 99.6264 /CMSY10 rf /Fr 134[35 40 1[34 1[25 33 32 32 36 34 43 62 21 37 29 24 41 34 34 4[37 6[48 41 58 1[41 48 41 43 53 56 45 1[56 68 48 60 39 31 58 55 45 52 58 50 53 53 37 2[35 1[20 20 18[46 4[33 1[46 1[42 1[31 40 36 41 31 35 42 41 1[25 1[35 31 29 31 2[45 11[{}65 66.4176 /CMMI8 rf /Fs 133[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 7[72 1[97 1[72 70 55 71 1[66 1[72 87 61 2[38 72 75 64 66 1[70 69 72 3[75 2[30 50 50 50 1[50 1[50 50 50 50 1[30 35 30 2[40 40 6[30 18[86 57 55 60 11[{}64 99.6264 /CMTI12 rf /Ft 133[50 59 59 1[59 62 44 44 46 59 62 56 62 93 31 59 34 31 62 56 34 51 62 50 62 54 7[85 3[86 78 62 84 1[77 1[88 106 67 88 1[42 88 88 70 74 86 81 80 85 6[31 56 56 56 56 56 56 56 56 56 56 1[31 37 3[44 28[62 12[{}59 99.6264 /CMBX12 rf /Fu 132[38 34 3[41 43 30 30 30 41 43 38 43 64 21 41 23 21 43 38 23 34 43 34 43 38 7[58 4[55 43 57 60 52 2[70 48 3[58 60 50 52 1[55 54 58 7[38 38 38 38 38 38 38 38 38 38 3[21 44[{}50 74.7198 /CMR9 rf /Fv 133[45 48 55 70 47 56 35 46 44 43 49 1[58 85 29 51 40 33 56 47 48 45 51 42 41 51 6[67 57 81 92 57 66 57 60 74 77 63 1[78 94 66 83 54 43 81 77 63 72 81 70 74 73 51 1[76 49 76 27 27 13[27 4[64 4[46 1[63 1[58 1[42 55 50 55 43 48 59 57 56 34 1[48 43 39 43 1[55 62 11[{}77 99.6264 /CMMI12 rf /Fw 133[50 3[61 61 59 46 60 63 56 63 61 74 51 63 42 30 61 64 53 56 62 59 58 61 7[81 4[78 61 80 84 74 2[99 68 3[81 85 71 74 1[78 77 81 20[32 44[{}39 99.6264 /CMCSC10 rf /Fx 128[49 49 2[49 43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 2[49 27 49 27 60 73 73 100 73 73 70 54 72 76 66 76 73 89 61 76 50 35 73 77 64 66 75 70 69 73 1[46 1[76 1[27 27 49 49 49 49 49 49 49 49 49 49 49 27 33 27 76 1[38 38 27 3[81 49 27 7[49 2[49 7[81 54 54 57 1[76 70 4[68 1[81 1[{}94 99.6264 /CMR12 rf /Fy 140[46 46 2[59 5[33 1[59 36 52 3[59 16[80 11[89 1[83 19[33 46[{}12 119.552 /CMR12 rf /Fz 169[119 3[117 3[119 1[99 4[125 1[108 122 115 113 119 65[{}10 172.188 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 1063 628 a Fz(AD)l(V)-18 b(ANCED)52 b(ALGEBRA)1423 959 y Fy(Prof.)37 b(Dr.)h(B.)h(P)m(areigis)1408 1203 y Fx(Win)m(ter)33 b(Semester)i(2001/02)1465 2683 y Fw(T)-9 b(able)38 b(of)g(Contents)100 2857 y Fx(1.)97 b(T)-8 b(ensor)34 b(Pro)s(ducts)f(and)g(F)-8 b(ree)32 b(Mo)s(dules)2085 b(3)100 2973 y(1.1.)97 b(Mo)s(dules)3145 b(3)100 3089 y(1.2.)97 b(T)-8 b(ensor)33 b(pro)s(ducts)h(I)2745 b(5)100 3206 y(1.3.)97 b(F)-8 b(ree)33 b(mo)s(dules)2940 b(6)100 3322 y(1.4.)97 b(T)-8 b(ensor)33 b(pro)s(ducts)h(I)s(I)2707 b(8)100 3438 y(1.5.)97 b(Bimo)s(dules)3057 b(9)100 3554 y(1.6.)97 b(Complexes)35 b(and)e(exact)g(sequences)2108 b(12)100 3671 y(2.)97 b(Algebras)33 b(and)g(Coalgebras)2469 b(15)100 3787 y(2.1.)97 b(Algebras)3082 b(15)100 3903 y(2.2.)97 b(T)-8 b(ensor)33 b(algebras)2789 b(17)100 4019 y(2.3.)97 b(Symmetric)34 b(algebras)2616 b(19)100 4135 y(2.4.)97 b(Exterior)33 b(algebras)2723 b(21)100 4252 y(2.5.)97 b(Left)32 b Fv(A)p Fx(-mo)s(dules)2794 b(23)100 4368 y(2.6.)97 b(Coalgebras)2987 b(23)100 4484 y(2.7.)97 b(Como)s(dules)2985 b(26)100 4600 y(3.)97 b(Pro)5 b(jectiv)m(e)35 b(Mo)s(dules)e(and)g(Generators)2017 b(30)100 4717 y(3.1.)97 b(Pro)s(ducts)33 b(and)g(copro)s(ducts)2384 b(30)100 4833 y(3.2.)97 b(Pro)5 b(jectiv)m(e)35 b(mo)s(dules)2639 b(34)100 4949 y(3.3.)97 b(Dual)32 b(basis)3010 b(36)100 5065 y(3.4.)97 b(Generators)2986 b(39)100 5182 y(4.)97 b(Categories)34 b(and)e(F)-8 b(unctors)2492 b(40)100 5298 y(4.1.)97 b(Categories)3009 b(40)100 5414 y(4.2.)97 b(F)-8 b(unctors)3083 b(42)100 5530 y(4.3.)97 b(Natural)32 b(T)-8 b(ransformations)2399 b(43)100 5646 y(5.)97 b(Represen)m(table) 35 b(and)e(Adjoin)m(t)f(F)-8 b(unctors,)34 b(the)f(Y)-8 b(oneda)32 b(Lemma)1097 b(46)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 0 -170 a Fu(2)1427 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)100 29 y Fx(5.1.)97 b(Represen)m(table)35 b(functors)2481 b(46)100 146 y(5.2.)97 b(The)33 b(Y)-8 b(oneda)33 b(Lemma)2591 b(49)100 262 y(5.3.)97 b(Adjoin)m(t)33 b(functors)2754 b(51)100 378 y(5.4.)97 b(Univ)m(ersal)34 b(problems)2638 b(52)100 494 y(6.)97 b(Limits)33 b(and)g(Colimits,)h (Pro)s(ducts)f(and)g(Equalizers)1574 b(55)100 611 y(6.1.)97 b(Limits)33 b(of)f(diagrams)2652 b(55)100 727 y(6.2.)97 b(Colimits)34 b(of)e(diagrams)2566 b(57)100 843 y(6.3.)97 b(Completeness)2875 b(58)100 959 y(6.4.)97 b(Adjoin)m(t)33 b(functors)g(and)g(limits)2293 b(59)100 1076 y(7.)97 b(The)34 b(Morita)e(Theorems)2590 b(60)100 1192 y(8.)97 b(Simple)34 b(and)e(Semisimple)j(rings)e(and)g(Mo)s(dules)1724 b(66)100 1308 y(8.1.)97 b(Simple)34 b(and)e(Semisimple)j(rings)2228 b(66)100 1424 y(8.2.)97 b(Injectiv)m(e)35 b(Mo)s(dules)2700 b(67)100 1540 y(8.3.)97 b(Simple)34 b(and)e(Semisimple)j(Mo)s(dules) 2077 b(70)100 1657 y(8.4.)97 b(No)s(etherian)33 b(Mo)s(dules)2592 b(73)100 1773 y(9.)97 b(Radical)33 b(and)f(So)s(cle)2767 b(76)100 1889 y(10.)97 b(Lo)s(calization)2961 b(81)100 2005 y(10.1.)97 b(Lo)s(cal)32 b(rings)2934 b(81)100 2122 y(10.2.)97 b(Lo)s(calization)2885 b(81)100 2238 y(11.)97 b(Monoidal)33 b(Categories)2605 b(87)100 2354 y(12.)97 b(Bialgebras)33 b(and)g(Hopf)f(Algebras)2205 b(92)100 2470 y(12.1.)97 b(Bialgebras)2961 b(92)100 2587 y(12.2.)97 b(Hopf)32 b(Algebras)2795 b(94)100 2703 y(13.)97 b(Quic)m(kies)35 b(in)d(Adv)-5 b(anced)34 b(Algebra)2145 b(101)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 1392 -170 a Fu(T)-6 b(ensor)26 b(pro)r(ducts)g(and)f(free)i(mo)r(dules)1329 b(3)958 29 y Fx(1.)48 b Fw(Tensor)38 b(Pr)n(oducts)f(and)h(Free)f(Modules)0 204 y Fx(1.1.)49 b Ft(Mo)s(dules.)0 379 y(De\014nition)29 b(1.1.)36 b Fx(Let)25 b Fv(R)h Fx(b)s(e)f(a)f(ring)h(\(alw)m(a)m(ys)h (asso)s(ciativ)m(e)h(with)e(unit)g(elemen)m(t\).)43 b(A)25 b Fs(left)i Fv(R)q Fs(-mo)-5 b(dule)3738 394 y Fr(R)3795 379 y Fv(M)0 495 y Fx(is)33 b(an)g(Ab)s(elian)g(group)f Fv(M)43 b Fx(\(with)33 b(comp)s(osition)h(written)f(as)g(addition\))g (together)f(with)i(an)e(op)s(eration)1358 654 y Fv(R)24 b Fq(\002)e Fv(M)39 b Fq(3)28 b Fx(\()p Fv(r)m(;)17 b(m)p Fx(\))28 b Fq(7!)f Fv(r)s(m)h Fq(2)g Fv(M)0 813 y Fx(suc)m(h)34 b(that)148 951 y(\(1\))42 b(\()p Fv(r)s(s)p Fx(\))p Fv(m)27 b Fx(=)h Fv(r)s Fx(\()p Fv(sm)p Fx(\),)148 1067 y(\(2\))42 b(\()p Fv(r)24 b Fx(+)e Fv(s)p Fx(\))p Fv(m)28 b Fx(=)g Fv(r)s(m)22 b Fx(+)g Fv(sm)p Fx(,)148 1183 y(\(3\))42 b Fv(r)s Fx(\()p Fv(m)22 b Fx(+)g Fv(m)690 1147 y Fp(0)713 1183 y Fx(\))28 b(=)f Fv(r)s(m)c Fx(+)f Fv(r)s(m)1267 1147 y Fp(0)1290 1183 y Fx(,)148 1299 y(\(4\))42 b(1)p Fv(m)27 b Fx(=)h Fv(m)0 1437 y Fx(for)k(all)h Fv(r)m(;)17 b(s)27 b Fq(2)h Fv(R)q Fx(,)33 b Fv(m;)17 b(m)886 1401 y Fp(0)937 1437 y Fq(2)28 b Fv(M)10 b Fx(.)0 1553 y(If)33 b Fv(R)g Fx(is)g(a)f(\014eld)i(then)f(a)f(\(left\))h Fv(R)q Fx(-mo)s(dule)g(is)g(a)f(\(called)h(a\))g(v)m(ector)g(space)h(o) m(v)m(er)g Fv(R)q Fx(.)0 1669 y(A)h Fs(homomorphism)g(of)i(left)g Fv(R)q Fs(-mo)-5 b(dules)34 b Fx(or)g(simply)j(an)e Fv(R)q Fs(-mo)-5 b(dule)36 b(homomorphism)d Fv(f)42 b Fx(:)3411 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Fx(is)g(ev)m(en,)i(and)1682 4527 y(0)p Fv(;)221 b Fx(if)32 b Fv(i)h Fx(is)g(o)s(dd.)1191 4753 y Fv(b)p Fx(\()p Fv(b)1311 4768 y Fr(i)1340 4753 y Fx(\))93 b(:=)1601 4582 y Fl(\()1682 4686 y Fv(b)1723 4702 y Fr(i)p Fp(\000)p Fk(1)p Fr(=)p Fk(2)1912 4686 y Fv(;)130 b Fx(if)33 b Fv(i)f Fx(is)h(o)s(dd,)g(and)1682 4826 y(0)p Fv(;)311 b Fx(if)33 b Fv(i)f Fx(is)h(ev)m(en.)315 5001 y(Sho)m(w)g Fv(pa)22 b Fx(+)g Fv(q)t(b)28 b Fx(=)g(id)1093 5016 y Fr(V)1154 5001 y Fx(,)33 b Fv(ap)27 b Fx(=)h Fv(bq)k Fx(=)c(id,)k Fv(aq)g Fx(=)c Fv(bp)g Fx(=)f(0.)414 5117 y(Sho)m(w)34 b(for)e Fv(R)c Fx(=)g(End)1202 5132 y Fr(K)1270 5117 y Fx(\()p Fv(V)22 b Fx(\))32 b(that)1668 5132 y Fr(R)1726 5117 y Fv(R)d Fx(=)e Fv(R)q(a)c Fq(\010)f Fv(R)q(b)33 b Fx(and)g Fv(R)2593 5132 y Fr(R)2679 5117 y Fx(=)27 b Fv(pR)c Fq(\010)g Fv(q)t(R)34 b Fx(holds.)148 5255 y(\(7\))42 b(Are)47 b Fq(f)p Fx(\(0)p Fv(;)17 b(:)g(:)g(:)e(;)i(a;)g(:) g(:)g(:)f(;)h Fx(0\))p Fq(j)p Fv(a)51 b Fq(2)h Fv(K)1559 5270 y Fr(n)1606 5255 y Fq(g)46 b Fx(and)h Fq(f)p Fx(\()p Fv(a;)17 b Fx(0)p Fv(;)g(:)g(:)g(:)f(;)h Fx(0\))p Fq(j)p Fv(a)51 b Fq(2)h Fv(K)2775 5270 y Fr(n)2822 5255 y Fq(g)46 b Fx(isomorphic)i(as)f Fv(M)3654 5270 y Fr(n)3701 5255 y Fx(\()p Fv(K)7 b Fx(\)-)315 5371 y(mo)s(dules?)148 5509 y(\(8\))42 b(F)-8 b(or)31 b(eac)m(h)j(mo)s(dule)f Fv(P)46 b Fx(there)33 b(is)g(a)g(mo)s(dule)g Fv(Q)f Fx(suc)m(h)j(that)d Fv(P)j Fq(\010)23 b Fv(Q)2779 5481 y Fq(\030)2780 5513 y Fx(=)2884 5509 y Fv(Q)p Fx(.)148 5646 y(\(9\))42 b(Whic)m(h)33 b(of)g(the)g(follo)m(wing)f(statemen)m(ts)j(is)e(correct?)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 1392 -170 a Fu(T)-6 b(ensor)26 b(pro)r(ducts)g(and)f(free)i(mo)r(dules)1329 b(5)347 29 y Fx(\(a\))41 b Fv(P)576 44 y Fk(1)638 29 y Fq(\010)22 b Fv(Q)28 b Fx(=)g Fv(P)1009 44 y Fk(2)1070 29 y Fq(\010)23 b Fv(Q)33 b Fx(=)-17 b Fq(\))32 b Fv(P)1534 44 y Fk(1)1601 29 y Fx(=)c Fv(P)1768 44 y Fk(2)1807 29 y Fx(?)342 146 y(\(b\))41 b Fv(P)576 161 y Fk(1)638 146 y Fq(\010)22 b Fv(Q)28 b Fx(=)g Fv(P)1009 161 y Fk(2)1070 146 y Fq(\010)23 b Fv(Q)33 b Fx(=)-17 b Fq(\))32 b Fv(P)1534 161 y Fk(1)1601 118 y Fq(\030)1602 150 y Fx(=)1706 146 y Fv(P)1769 161 y Fk(2)1809 146 y Fx(?)352 262 y(\(c\))42 b Fv(P)576 277 y Fk(1)638 262 y Fq(\010)22 b Fv(Q)842 234 y Fq(\030)843 266 y Fx(=)947 262 y Fv(P)1010 277 y Fk(2)1072 262 y Fq(\010)g Fv(Q)33 b Fx(=)-17 b Fq(\))33 b Fv(P)1536 277 y Fk(1)1603 234 y Fq(\030)1603 266 y Fx(=)1708 262 y Fv(P)1771 277 y Fk(2)1810 262 y Fx(?)100 400 y(\(10\))41 b Fn(Z)p Fv(=)p Fx(\(2\))22 b Fq(\010)g Fn(Z)p Fv(=)p Fx(\(6\))g Fq(\010)h Fn(Z)p Fv(=)p Fx(\(6\))f Fq(\010)g Fv(:)17 b(:)g(:)1542 372 y Fq(\030)1542 404 y Fx(=)1647 400 y Fn(Z)p Fv(=)p Fx(\(6\))22 b Fq(\010)g Fn(Z)p Fv(=)p Fx(\(6\))g Fq(\010)h Fn(Z)p Fv(=)p Fx(\(6\))f Fq(\010)h Fv(:)17 b(:)g(:)o Fx(.)100 537 y(\(11\))41 b Fn(Z)p Fv(=)p Fx(\(2\))22 b Fq(\010)g Fn(Z)p Fv(=)p Fx(\(4\))g Fq(\010)h Fn(Z)p Fv(=)p Fx(\(4\))f Fq(\010)g Fv(:)17 b(:)g(:)28 b Fq(6)1542 510 y(\030)1543 541 y Fx(=)1647 537 y Fn(Z)p Fv(=)p Fx(\(4\))22 b Fq(\010)g Fn(Z)p Fv(=)p Fx(\(4\))g Fq(\010)h Fn(Z)p Fv(=)p Fx(\(4\))f Fq(\010)h Fv(:)17 b(:)g(:)o Fx(.)100 675 y(\(12\))41 b(Find)e(t)m(w)m(o)h(Ab)s(elian)g (groups)g Fv(P)53 b Fx(and)39 b Fv(Q)p Fx(,)j(suc)m(h)f(that)e Fv(P)53 b Fx(is)40 b(isomorphic)g(to)f(a)h(subgroup)g(of)f Fv(Q)315 791 y Fx(and)32 b Fv(Q)h Fx(is)g(isomorphic)h(to)e(a)g (subgroup)i(of)e Fv(P)46 b Fx(and)32 b Fv(P)41 b Fq(6)2343 764 y(\030)2344 795 y Fx(=)2449 791 y Fv(Q)p Fx(.)0 970 y(1.2.)49 b Ft(T)-9 b(ensor)37 b(pro)s(ducts)h(I.)0 1150 y(De\014nition)46 b(and)h(Remark)f(1.4.)f Fx(Let)40 b Fv(M)1684 1165 y Fr(R)1781 1150 y Fx(and)1978 1165 y Fr(R)2036 1150 y Fv(N)50 b Fx(b)s(e)40 b Fv(R)q Fx(-mo)s(dules,)i(and)e (let)f Fv(A)h Fx(b)s(e)g(an)f(Ab)s(elian)0 1266 y(group.)k(A)33 b(map)g Fv(f)38 b Fx(:)28 b Fv(M)33 b Fq(\002)22 b Fv(N)39 b Fq(\000)-60 b(!)27 b Fv(A)33 b Fx(is)g(called)h Fv(R)q Fs(-biline)-5 b(ar)31 b Fx(if)148 1407 y(\(1\))42 b Fv(f)11 b Fx(\()p Fv(m)22 b Fx(+)g Fv(m)702 1371 y Fp(0)725 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Fv(M)5 b(;)17 b(N)10 b Fx(;)17 b Fv(A)p Fx(\))33 b(is)g(an)f(Ab)s(elian)h(group)g(with)g(\()p Fv(f)g Fx(+)22 b Fv(g)t Fx(\)\()p Fv(m;)17 b(n)p Fx(\))27 b(:=)g Fv(f)11 b Fx(\()p Fv(m;)17 b(n)p Fx(\))22 b(+)g Fv(g)t Fx(\()p Fv(m;)17 b(n)p Fx(\).)0 2192 y Ft(De\014nition)38 b(1.5.)j Fx(Let)33 b Fv(M)1007 2207 y Fr(R)1097 2192 y Fx(and)1286 2207 y Fr(R)1344 2192 y Fv(N)42 b Fx(b)s(e)33 b Fv(R)q Fx(-mo)s(dules.)44 b(An)32 b(Ab)s(elian)h(group)f Fv(M)g Fq(\012)3126 2207 y Fr(R)3205 2192 y Fv(N)43 b Fx(together)32 b(with)0 2308 y(an)g Fv(R)q Fx(-bilinear)h(map)1063 2433 y Fq(\012)28 b Fx(:)g Fv(M)k Fq(\002)23 b Fv(N)38 b Fq(3)28 b Fx(\()p Fv(m;)17 b(n)p Fx(\))28 b Fq(7!)g Fv(m)22 b Fq(\012)h Fv(n)k Fq(2)i Fv(M)k Fq(\012)2669 2448 y Fr(R)2749 2433 y Fv(N)0 2578 y Fx(is)39 b(called)h(a)e Fs(tensor)i(pr)-5 b(o)g(duct)41 b(of)78 b Fv(M)50 b Fs(and)78 b Fv(N)49 b Fs(over)38 b Fv(R)i Fx(if)e(for)g(eac)m(h)i(Ab)s(elian)f (group)g Fv(A)f Fx(and)h(for)f(eac)m(h)0 2694 y Fv(R)q Fx(-bilinear)f(map)h Fv(f)46 b Fx(:)36 b Fv(M)g Fq(\002)26 b Fv(N)46 b Fq(\000)-60 b(!)35 b Fv(A)j Fx(there)f(exists)i(a)e(unique) i(group)e(homomorphism)h Fv(g)h Fx(:)d Fv(M)g Fq(\012)3728 2709 y Fr(R)3811 2694 y Fv(N)0 2811 y Fq(\000)-60 b(!)28 b Fv(A)k Fx(suc)m(h)i(that)f(the)g(diagram)1534 2959 y Fv(M)g Fq(\002)23 b Fv(N)154 b(M)33 b Fq(\012)2197 2974 y Fr(R)2277 2959 y Fv(N)p 1878 2934 86 4 v 1881 2932 a Fj(-)1882 2905 y Fq(\012)1816 3201 y Fv(f)1760 3084 y Fj(@)1843 3167 y(@)1926 3250 y(@)2009 3333 y(@)2028 3352 y(@)-83 b(R)2143 3454 y Fv(A)p 2177 3352 4 351 v 2179 3352 a Fj(?)2218 3188 y Fv(g)0 3565 y Fx(comm)m(utes.)45 b(The)33 b(elemen)m(ts)h(of)d Fv(M)g Fq(\012)1406 3580 y Fr(R)1485 3565 y Fv(N)42 b Fx(are)32 b(called)h Fs(tensors)p Fx(,)e(the)h(elemen)m(ts)i(of)e(the)g(form)f Fv(m)21 b Fq(\012)g Fv(n)32 b Fx(are)0 3682 y(called)h Fs(de)-5 b(c)g(omp)g(osable)33 b(tensors)p Fx(.)0 3798 y Fs(Warning:)47 b Fx(If)34 b(y)m(ou)h(w)m(an)m(t)h(to)e(de\014ne)i(a)e(homomorphism)i Fv(f)41 b Fx(:)31 b Fv(M)k Fq(\012)2508 3813 y Fr(R)2589 3798 y Fv(N)42 b Fq(\000)-60 b(!)31 b Fv(A)j Fx(with)h(a)g(tensor)g (pro)s(duct)0 3914 y(as)e(domain)g(y)m(ou)g Fs(must)g Fx(de\014ne)g(it)g(b)m(y)g(giving)g(an)g Fv(R)q Fx(-bilinear)g(map)g (de\014ned)h(on)e Fv(M)h Fq(\002)23 b Fv(N)10 b Fx(.)0 4093 y Ft(Prop)s(osition)37 b(1.6.)42 b Fs(A)35 b(tensor)f(pr)-5 b(o)g(duct)35 b Fx(\()p Fv(M)d Fq(\012)1811 4108 y Fr(R)1891 4093 y Fv(N)5 b(;)17 b Fq(\012)p Fx(\))34 b Fs(de\014ne)-5 b(d)34 b(by)h Fv(M)2717 4108 y Fr(R)2809 4093 y Fs(and)2998 4108 y Fr(R)3056 4093 y Fv(N)45 b Fs(is)34 b(unique)h(up)f(to)h(a)0 4210 y(unique)g(isomorphism.)0 4389 y(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b(\()p Fv(M)g Fq(\012)715 4404 y Fr(R)795 4389 y Fv(N)5 b(;)17 b Fq(\012)p Fx(\))33 b(and)g(\()p Fv(M)f Fo(\002)1501 4404 y Fr(R)1582 4389 y Fv(N)5 b(;)17 b Fo(\002)p Fx(\))32 b(b)s(e)h(tensor)g(pro)s(ducts.)45 b(Then)1792 4539 y Fv(M)33 b Fq(\002)23 b Fv(N)1323 4778 y Fq(\012)1730 4619 y Fj(\010)1647 4661 y(\010)1564 4702 y(\010)1481 4744 y(\010)1398 4785 y(\010)1315 4827 y(\010)1232 4868 y(\010)1149 4910 y(\010)1112 4929 y(\010)-83 b(\031)1767 4787 y Fo(\002)1799 4661 y Fj(\000)1716 4744 y(\000)1633 4827 y(\000)1550 4910 y(\000)1531 4929 y(\000)g(\011)2056 4778 y Fq(\012)2018 4661 y Fj(@)2101 4744 y(@)2184 4827 y(@)2267 4910 y(@)2286 4929 y(@)g(R)2500 4787 y Fo(\002)2086 4619 y Fj(H)2170 4661 y(H)2253 4702 y(H)2336 4744 y(H)2419 4785 y(H)2502 4827 y(H)2585 4868 y(H)2668 4910 y(H)2705 4929 y(H)g(j)789 5023 y Fv(M)33 b Fq(\012)993 5038 y Fr(R)1073 5023 y Fv(N)125 b(M)33 b Fo(\002)1480 5038 y Fr(R)1560 5023 y Fv(N)p 1191 4999 57 4 v 1165 4997 a Fj(-)1191 4978 y Fv(h)p 1678 4999 545 4 v 2140 4997 a Fj(-)1923 4978 y Fv(k)2251 5023 y(M)g Fq(\012)2455 5038 y Fr(R)2535 5023 y Fv(N)126 b(M)32 b Fo(\002)2942 5038 y Fr(R)3023 5023 y Fv(N)p 2653 4999 57 4 v 2627 4997 a Fj(-)2653 4978 y Fv(h)0 5174 y Fx(implies)i Fv(k)d Fx(=)c Fv(h)572 5138 y Fp(\000)p Fk(1)667 5174 y Fx(.)3129 b Fo(\003)0 5351 y Fx(Because)34 b(of)e(this)h(fact)g(w)m(e)g(will)h (henceforth)f(talk)g(ab)s(out)f Fs(the)h Fx(tensor)g(pro)s(duct)g(of)f Fv(M)43 b Fx(and)33 b Fv(N)43 b Fx(o)m(v)m(er)34 b Fv(R)q Fx(.)0 5530 y Ft(Prop)s(osition)c(1.7.)36 b Fx(\(Rules)27 b(of)e(computation)h(in)g(a)g(tensor)g(pro)s(duct\))j Fs(L)-5 b(et)29 b Fx(\()p Fv(M)19 b Fq(\012)3031 5545 y Fr(R)3097 5530 y Fv(N)5 b(;)17 b Fq(\012)p Fx(\))29 b Fs(b)-5 b(e)29 b(the)f(tensor)0 5646 y(pr)-5 b(o)g(duct.)45 b(Then)34 b(we)g(have)g(for)h(al)5 b(l)35 b Fv(r)30 b Fq(2)e Fv(R)q Fs(,)35 b Fv(m;)17 b(m)1825 5610 y Fp(0)1876 5646 y Fq(2)28 b Fv(M)10 b Fs(,)36 b Fv(n;)17 b(n)2300 5610 y Fp(0)2351 5646 y Fq(2)28 b Fv(N)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 0 -170 a Fu(6)1427 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)148 29 y Fx(\(1\))42 b Fv(M)32 b Fq(\012)518 44 y Fr(R)599 29 y Fv(N)38 b Fx(=)28 b Fq(f)869 -45 y Fl(P)974 59 y Fr(i)1018 29 y Fv(m)1103 44 y Fr(i)1154 29 y Fq(\012)23 b Fv(n)1312 44 y Fr(i)1375 29 y Fq(j)34 b Fv(m)1522 44 y Fr(i)1579 29 y Fq(2)28 b Fv(M)5 b(;)17 b(n)1874 44 y Fr(i)1930 29 y Fq(2)28 b Fv(N)10 b Fq(g)p Fv(;)148 146 y Fx(\(2\))42 b(\()p Fv(m)22 b Fx(+)g Fv(m)643 110 y Fp(0)667 146 y Fx(\))g Fq(\012)g Fv(n)28 b Fx(=)g Fv(m)22 b Fq(\012)h Fv(n)f Fx(+)g Fv(m)1486 110 y Fp(0)1532 146 y Fq(\012)h Fv(n;)148 262 y Fx(\(3\))42 b Fv(m)22 b Fq(\012)h Fx(\()p Fv(n)f Fx(+)g Fv(n)796 226 y Fp(0)819 262 y Fx(\))28 b(=)g Fv(m)22 b Fq(\012)h Fv(n)f Fx(+)g Fv(m)h Fq(\012)f Fv(n)1639 226 y Fp(0)1663 262 y Fv(;)148 378 y Fx(\(4\))42 b Fv(mr)24 b Fq(\012)d Fv(n)28 b Fx(=)g Fv(m)21 b Fq(\012)h Fv(r)s(n)34 b Fs(\(observe)g(in)g(p)-5 b(articular,)34 b(that)h Fq(\012)28 b Fx(:)g Fv(M)k Fq(\002)21 b Fv(N)39 b Fq(\000)-58 b(!)28 b Fv(M)k Fq(\012)21 b Fv(N)45 b Fs(is)34 b(not)h(inje)-5 b(ctive)315 494 y(in)34 b(gener)-5 b(al\),)148 611 y Fx(\(5\))42 b Fs(if)i Fv(f)57 b Fx(:)46 b Fv(M)40 b Fq(\002)30 b Fv(N)56 b Fq(\000)-57 b(!)46 b Fv(A)e Fs(is)h(an)f Fv(R)q Fs(-biline)-5 b(ar)44 b(map)g(and)g Fv(g)49 b Fx(:)d Fv(M)40 b Fq(\012)2792 626 y Fr(R)2880 611 y Fv(N)56 b Fq(\000)-57 b(!)46 b Fv(A)f Fs(is)f(the)h(induc)-5 b(e)g(d)315 727 y(homomorphism,)32 b(then)1514 893 y Fv(g)t Fx(\()p Fv(m)22 b Fq(\012)g Fv(n)p Fx(\))28 b(=)g Fv(f)11 b Fx(\()p Fv(m;)17 b(n)p Fx(\))p Fv(:)0 1096 y Fs(Pr)-5 b(o)g(of.)41 b Fx(\(1\))33 b(Let)g Fv(B)h Fx(:=)29 b Fq(h)p Fv(m)23 b Fq(\012)g Fv(n)p Fq(i)29 b(\022)g Fv(M)k Fq(\012)1554 1111 y Fr(R)1635 1096 y Fv(N)44 b Fx(denote)34 b(the)g(subgroup)g(of)e Fv(M)i Fq(\012)2981 1111 y Fr(R)3062 1096 y Fv(N)43 b Fx(generated)34 b(b)m(y)h(the)0 1212 y(decomp)s(osable)f(tensors)f Fv(m)22 b Fq(\012)g Fv(n)p Fx(.)44 b(Let)32 b Fv(j)i Fx(:)27 b Fv(B)33 b Fq(\000)-59 b(!)27 b Fv(M)32 b Fq(\012)2044 1227 y Fr(R)2124 1212 y Fv(N)43 b Fx(b)s(e)32 b(the)h(em)m(b)s(edding)h (homomorphism.)45 b(W)-8 b(e)0 1328 y(get)33 b(an)f(induced)i(map)f Fq(\012)955 1292 y Fp(0)1006 1328 y Fx(:)28 b Fv(M)33 b Fq(\002)23 b Fv(N)38 b Fq(\000)-60 b(!)28 b Fv(B)5 b Fx(.)43 b(The)34 b(follo)m(wing)f(diagram)1290 1580 y Fv(M)g Fq(\002)23 b Fv(N)301 b(B)p 1634 1551 234 4 v 1785 1549 a Fj(-)1700 1523 y Fq(\012)1777 1486 y Fp(0)2236 1576 y Fv(M)33 b Fq(\012)2440 1591 y Fr(R)2521 1576 y Fv(N)p 2004 1551 204 4 v 2125 1549 a Fj(-)2082 1511 y Fv(j)1896 2064 y(B)266 b(M)33 b Fq(\012)2440 2079 y Fr(R)2521 2064 y Fv(N)p 2004 2039 V 2125 2037 a Fj(-)2082 1999 y Fv(j)1530 1826 y Fq(\012)1607 1790 y Fp(0)1516 1701 y Fj(@)1599 1784 y(@)1682 1867 y(@)1765 1950 y(@)1784 1969 y(@)-83 b(R)p 2421 1969 4 351 v 556 w(?)2462 1816 y Fv(j)6 b(p)2240 1805 y(p)2272 1701 y Fj(\000)2189 1784 y(\000)2106 1867 y(\000)2023 1950 y(\000)2004 1969 y(\000)-83 b(\011)0 2250 y Fx(induces)35 b(a)f(unique)h Fv(p)e Fx(with)h Fv(p)23 b Fq(\016)g Fv(j)29 b Fq(\016)22 b(\012)1417 2214 y Fp(0)1470 2250 y Fx(=)30 b Fv(p)22 b Fq(\016)h(\012)30 b Fx(=)f Fq(\012)2009 2214 y Fp(0)2066 2250 y Fx(since)35 b Fq(\012)2383 2214 y Fp(0)2441 2250 y Fx(is)f Fv(R)q Fx(-bilinear.)47 b(Because)35 b(of)e Fv(j)6 b(p)23 b Fq(\016)f(\012)30 b Fx(=)0 2366 y 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Fx(\))35 b Fs(and)f Fv(g)d Fq(2)d Fx(Hom)2461 5347 y Fr(R)2519 5332 y Fx(\()p Fv(:N)5 b(;)17 b(:N)2826 5291 y Fp(0)2849 5332 y Fx(\))p Fv(:)0 5489 y Fs(Then)34 b(ther)-5 b(e)35 b(is)g(a)f(unique)h(homomorphism)1190 5646 y Fv(f)e Fq(\012)1348 5661 y Fr(R)1428 5646 y Fv(g)e Fq(2)d Fx(Hom)q(\()p Fv(M)k Fq(\012)2045 5661 y Fr(R)2126 5646 y Fv(N)5 b(;)17 b(M)2357 5605 y Fp(0)2403 5646 y Fq(\012)2480 5661 y Fr(R)2560 5646 y Fv(N)2648 5605 y Fp(0)2672 5646 y Fx(\))p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1392 -170 a Fu(T)-6 b(ensor)26 b(pro)r(ducts)g(and)f(free)i(mo)r(dules)1329 b(9)0 29 y Fs(such)35 b(that)g Fv(f)e Fq(\012)579 44 y Fr(R)659 29 y Fv(g)t Fx(\()p Fv(m)22 b Fq(\012)g Fv(n)p Fx(\))28 b(=)g Fv(f)11 b Fx(\()p Fv(m)p Fx(\))22 b Fq(\012)h Fv(g)t Fx(\()p Fv(n)p Fx(\))p Fs(,)34 b(i.e.)44 b(the)35 b(fol)5 b(lowing)33 b(diagr)-5 b(am)34 b(c)-5 b(ommutes)1448 706 y Fv(M)1552 670 y Fp(0)1598 706 y Fq(\002)23 b Fv(N)1786 670 y Fp(0)2004 706 y Fv(M)2108 670 y Fp(0)2154 706 y Fq(\012)2231 721 y Fr(R)2312 706 y Fv(N)2400 670 y Fp(0)p 1838 679 137 4 v 1892 677 a Fj(-)1868 754 y Fq(\012)1471 216 y Fv(M)33 b Fq(\002)23 b Fv(N)251 b(M)33 b Fq(\012)2231 231 y Fr(R)2312 216 y Fv(N)p 1815 191 184 4 v 1916 189 a Fj(-)1868 161 y Fq(\012)p 1627 608 4 351 v 1629 608 a Fj(?)1359 458 y Fv(f)g Fq(\002)22 b Fv(g)p 2212 608 V 2214 608 a Fj(?)2253 458 y Fv(f)33 b Fq(\012)2411 473 y Fr(R)2491 458 y Fv(g)0 885 y Fs(Pr)-5 b(o)g(of.)41 b Fq(\012)23 b(\016)f Fx(\()p Fv(f)33 b Fq(\002)22 b Fv(g)t Fx(\))32 b(is)h(bilinear.)2570 b Fo(\003)0 1041 y Ft(Notation)37 b(1.16.)42 b Fx(W)-8 b(e)33 b(often)g(write)g Fv(f)g Fq(\012)1566 1056 y Fr(R)1646 1041 y Fv(N)38 b Fx(:=)28 b Fv(f)33 b Fq(\012)2051 1056 y Fr(R)2131 1041 y Fx(1)2180 1056 y Fr(N)2280 1041 y Fx(and)f Fv(M)h Fq(\012)2673 1056 y Fr(R)2754 1041 y Fv(g)e Fx(:=)c(1)3011 1056 y Fr(M)3112 1041 y Fq(\012)3189 1056 y Fr(R)3269 1041 y Fv(g)t Fx(.)0 1158 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)g (rule)g(of)f(computation:)744 1315 y Fv(f)h Fq(\012)902 1330 y Fr(R)982 1315 y Fv(g)e Fx(=)d(\()p Fv(f)33 b Fq(\012)1360 1330 y Fr(R)1440 1315 y Fv(N)1528 1273 y Fp(0)1552 1315 y Fx(\))22 b Fq(\016)g Fx(\()p Fv(M)33 b Fq(\012)1926 1330 y Fr(R)2006 1315 y Fv(g)t Fx(\))27 b(=)g(\()p Fv(M)2367 1273 y Fp(0)2414 1315 y Fq(\012)2491 1330 y Fr(R)2571 1315 y Fv(g)t Fx(\))21 b Fq(\016)h Fx(\()p Fv(f)33 b Fq(\012)2949 1330 y Fr(R)3029 1315 y Fv(N)10 b Fx(\))0 1471 y(since)34 b Fv(f)f Fq(\002)22 b Fv(g)31 b Fx(=)d(\()p Fv(f)33 b Fq(\002)22 b Fv(N)907 1435 y Fp(0)931 1471 y Fx(\))g Fq(\016)g Fx(\()p Fv(M)33 b Fq(\002)23 b Fv(g)t Fx(\))k(=)g(\()p Fv(M)1689 1435 y Fp(0)1735 1471 y Fq(\002)c Fv(g)t Fx(\))e Fq(\016)h Fx(\()p Fv(f)33 b Fq(\002)23 b Fv(N)10 b Fx(\).)0 1628 y(1.5.)49 b Ft(Bimo)s(dules.)0 1784 y(De\014nition)27 b(1.17.)35 b Fx(Let)23 b Fv(R)q Fx(,)j Fv(S)j Fx(b)s(e)23 b(rings)h(and)f(let)h Fv(M)34 b Fx(b)s(e)23 b(a)g(left)g Fv(R)q Fx(-mo)s(dule)h(and)f(a)g(righ)m(t)g Fv(S)6 b Fx(-mo)s(dule.)40 b Fv(M)34 b Fx(is)0 1900 y(called)g(an)f Fv(R)q Fx(-)p Fv(S)6 b Fx(-)p Fs(bimo)-5 b(dule)31 b Fx(if)i(\()p Fv(r)s(m)p Fx(\))p Fv(s)c Fx(=)f Fv(r)s Fx(\()p Fv(ms)p Fx(\).)46 b(W)-8 b(e)33 b(de\014ne)i(Hom)2481 1915 y Fr(R)p Fx(-)p Fr(S)2618 1900 y Fx(\()p Fv(:M)5 b(:;)17 b(:N)5 b(:)p Fx(\))29 b(:=)f(Hom)3392 1915 y Fr(R)3449 1900 y Fx(\()p Fv(:M)5 b(;)17 b(:N)10 b Fx(\))24 b Fq(\\)0 2016 y Fx(Hom)203 2031 y Fr(S)254 2016 y Fx(\()p Fv(M)5 b(:;)17 b(N)5 b(:)p Fx(\).)0 2172 y Ft(Remark)49 b(1.18.)d Fx(Let)c Fv(M)990 2187 y Fr(S)1083 2172 y Fx(b)s(e)h(a)e (righ)m(t)h Fv(S)6 b Fx(-mo)s(dule)42 b(and)g(let)g Fv(R)30 b Fq(\002)f Fv(M)54 b Fq(\000)-59 b(!)43 b Fv(M)52 b Fx(b)s(e)43 b(a)e(map.)72 b Fv(M)53 b Fx(is)42 b(an)0 2288 y Fv(R)q Fx(-)p Fv(S)6 b Fx(-bimo)s(dule)32 b(if)h(and)f(only)h (if)148 2425 y(\(1\))42 b Fq(8)p Fv(r)30 b Fq(2)e Fv(R)h Fx(:)f(\()p Fv(M)38 b Fq(3)28 b Fv(m)g Fq(7!)g Fv(r)s(m)f Fq(2)h Fv(M)10 b Fx(\))29 b Fq(2)f Fx(Hom)1922 2440 y Fr(S)1973 2425 y Fx(\()p Fv(M)5 b(:;)17 b(M)5 b(:)p Fx(\),)148 2541 y(\(2\))42 b Fq(8)p Fv(r)m(;)17 b(r)502 2505 y Fp(0)552 2541 y Fq(2)29 b Fv(R)q(;)17 b(m)27 b Fq(2)h Fv(M)39 b Fx(:)28 b(\()p Fv(r)c Fx(+)e Fv(r)1411 2505 y Fp(0)1434 2541 y Fx(\))p Fv(m)28 b Fx(=)g Fv(r)s(m)22 b Fx(+)g Fv(r)1988 2505 y Fp(0)2011 2541 y Fv(m)p Fx(,)148 2658 y(\(3\))42 b Fq(8)p Fv(r)m(;)17 b(r)502 2621 y Fp(0)552 2658 y Fq(2)29 b Fv(R)q(;)17 b(m)27 b Fq(2)h Fv(M)39 b Fx(:)28 b(\()p Fv(r)s(r)1292 2621 y Fp(0)1314 2658 y Fx(\))p Fv(m)g Fx(=)g Fv(r)s Fx(\()p Fv(r)1701 2621 y Fp(0)1723 2658 y Fv(m)p Fx(\),)148 2774 y(\(4\))42 b Fq(8)p Fv(m)28 b Fq(2)g Fv(M)38 b Fx(:)28 b(1)p Fv(m)g Fx(=)g Fv(m:)0 2930 y Ft(Lemma)38 b(1.19.)j Fs(L)-5 b(et)834 2945 y Fr(R)891 2930 y Fv(M)985 2945 y Fr(S)1070 2930 y Fs(and)1258 2945 y Fr(S)1309 2930 y Fv(N)1387 2945 y Fr(T)1476 2930 y Fs(b)g(e)33 b(bimo)-5 b(dules.)44 b(Then)2332 2945 y Fr(R)2390 2930 y Fx(\()p Fv(M)30 b Fq(\012)2629 2945 y Fr(S)2700 2930 y Fv(N)10 b Fx(\))2826 2945 y Fr(T)2915 2930 y Fs(is)34 b(a)f(bimo)-5 b(dule)33 b(by)h Fv(r)s Fx(\()p Fv(m)20 b Fq(\012)0 3046 y Fv(n)p Fx(\))28 b(:=)f Fv(r)s(m)c Fq(\012)f Fv(n)35 b Fs(and)g Fx(\()p Fv(m)22 b Fq(\012)h Fv(n)p Fx(\))p Fv(t)28 b Fx(:=)f Fv(m)c Fq(\012)f Fv(nt)p Fs(.)0 3202 y(Pr)-5 b(o)g(of.)41 b Fx(Clearly)27 b(w)m(e)g(ha)m(v)m(e)g(that)f(\()p Fv(r)11 b Fq(\012)1360 3217 y Fr(S)1420 3202 y Fx(id\)\()p Fv(m)e Fq(\012)g Fv(n)p Fx(\))28 b(=)g Fv(r)s(m)9 b Fq(\012)g Fv(n)27 b Fx(=)h Fv(r)s Fx(\()p Fv(m)9 b Fq(\012)g Fv(n)p Fx(\))25 b(is)i(a)e(homomorphism.)43 b(Then)0 3318 y(\(2\)-\(4\))31 b(hold.)43 b(Th)m(us)34 b Fv(M)e Fq(\012)1018 3333 y Fr(S)1090 3318 y Fv(N)42 b Fx(is)33 b(a)e(left)i Fv(R)q Fx(-mo)s(dule.)43 b(Similarly)33 b(it)f(is)h(a)e(righ)m(t)h Fv(T)14 b Fx(-mo)s(dule.)44 b(Finally)32 b(w)m(e)0 3435 y(ha)m(v)m(e)i Fv(r)s Fx(\(\()p Fv(m)22 b Fq(\012)h Fv(n)p Fx(\))p Fv(t)p Fx(\))28 b(=)f Fv(r)s Fx(\()p Fv(m)22 b Fq(\012)h Fv(nt)p Fx(\))28 b(=)f Fv(r)s(m)c Fq(\012)f Fv(nt)28 b Fx(=)g(\()p Fv(r)s(m)22 b Fq(\012)h Fv(n)p Fx(\))p Fv(t)28 b Fx(=)f(\()p Fv(r)s Fx(\()p Fv(m)22 b Fq(\012)h Fv(n)p Fx(\)\))p Fv(t)p Fx(.)855 b Fo(\003)0 3591 y Ft(Corollary)73 b(1.20.)56 b Fs(Given)61 b(bimo)-5 b(dules)1603 3606 y Fr(R)1661 3591 y Fv(M)1755 3606 y Fr(S)1806 3591 y Fs(,)1904 3606 y Fr(S)1955 3591 y Fv(N)2033 3606 y Fr(T)2088 3591 y Fs(,)2186 3606 y Fr(R)2243 3591 y Fv(M)2347 3555 y Fp(0)2337 3616 y Fr(S)2389 3591 y Fs(,)2487 3606 y Fr(S)2537 3591 y Fv(N)2625 3555 y Fp(0)2615 3616 y Fr(T)2732 3591 y Fs(and)61 b(homomorphisms)e Fv(f)88 b Fq(2)0 3707 y Fx(Hom)203 3722 y Fr(R)p Fs(-)q Fr(S)342 3707 y Fx(\()p Fv(:M)5 b(:;)17 b(:M)708 3671 y Fp(0)733 3707 y Fv(:)p Fx(\))64 b Fs(and)g Fv(g)85 b Fq(2)e Fx(Hom)1565 3722 y Fr(S)t Fs(-)o Fr(T)1701 3707 y Fx(\()p Fv(:N)5 b(:;)17 b(:N)2035 3671 y Fp(0)2059 3707 y Fv(:)p Fx(\))p Fs(.)133 b(Then)63 b(we)h(have)g Fv(f)54 b Fq(\012)3177 3722 y Fr(S)3273 3707 y Fv(g)85 b Fq(2)d Fx(Hom)3757 3722 y Fr(R)p Fs(-)p Fr(T)0 3823 y Fx(\()p Fv(:M)33 b Fq(\012)269 3838 y Fr(S)342 3823 y Fv(N)5 b(:;)17 b(:M)627 3787 y Fp(0)674 3823 y Fq(\012)751 3838 y Fr(S)824 3823 y Fv(N)912 3787 y Fp(0)936 3823 y Fv(:)p Fx(\))p Fv(:)0 3979 y Fs(Pr)-5 b(o)g(of.)41 b Fv(f)33 b Fq(\012)456 3994 y Fr(S)529 3979 y Fv(g)t Fx(\()p Fv(r)s(m)22 b Fq(\012)g Fv(nt)p Fx(\))28 b(=)g Fv(f)11 b Fx(\()p Fv(r)s(m)p Fx(\))22 b Fq(\012)g Fv(g)t Fx(\()p Fv(nt)p Fx(\))28 b(=)f Fv(r)s Fx(\()p Fv(f)33 b Fq(\012)2116 3994 y Fr(S)2189 3979 y Fv(g)t Fx(\)\()p Fv(m)22 b Fq(\012)g Fv(n)p Fx(\))p Fv(t:)1143 b Fo(\003)0 4135 y Ft(Remark)38 b(1.21.)k Fx(Unless)34 b(otherwise)g(de\014ned)g Fn(K)f Fx(will)g(alw)m(a)m(ys)i (b)s(e)d(a)h(comm)m(utativ)m(e)h(ring.)0 4252 y(Ev)m(ery)k(mo)s(dule)e Fv(M)46 b Fx(o)m(v)m(er)37 b(the)f(comm)m(utativ)m(e)i(ring)e Fn(K)g Fx(and)f(in)h(particular)g(ev)m(ery)i(v)m(ector)f(space)f(o)m(v) m(er)h(a)0 4368 y(\014eld)31 b Fn(K)f Fx(is)g(a)g Fn(K)p Fx(-)p Fn(K)p Fx(-bimo)s(dule)g(b)m(y)h Fv(\025m)d Fx(=)f Fv(m\025)p Fx(.)43 b(Observ)m(e)32 b(that)d(there)i(are)f Fn(K)p Fx(-)p Fn(K)p Fx(-bimo)s(dules)h(that)e(do)h(not)0 4484 y(satisfy)36 b Fv(\025m)c Fx(=)f Fv(m\025)p Fx(.)51 b(T)-8 b(ak)m(e)36 b(for)e(example)j(an)d(automorphism)i Fv(\013)c Fx(:)g Fn(K)g Fq(\000)-59 b(!)31 b Fn(K)k Fx(and)g(a)g(left)g Fn(K)p Fx(-mo)s(dule)g Fv(M)0 4600 y Fx(and)e(de\014ne)g Fv(m\025)28 b Fx(:=)g Fv(\013)q Fx(\()p Fv(\025)p Fx(\))p Fv(m)p Fx(.)43 b(Then)34 b Fv(M)43 b Fx(is)33 b(suc)m(h)h(a)f Fn(K)p Fx(-)p Fn(K)p Fx(-bimo)s(dule.)0 4717 y(The)i(tensor)f(pro)s (duct)g Fv(M)g Fq(\012)1068 4732 y Fm(K)1147 4717 y Fv(N)45 b Fx(of)33 b(t)m(w)m(o)i Fn(K)p Fx(-)p Fn(K)p Fx(-bimo)s(dules)g Fv(M)44 b Fx(and)34 b Fv(N)45 b Fx(is)34 b(again)f(a)h Fn(K)p Fx(-)p Fn(K)p Fx(-bimo)s(dule.)48 b(If)0 4833 y(w)m(e)29 b(ha)m(v)m(e,)g(ho)m(w)m(ev)m(er,)i Fn(K)p Fx(-)p Fn(K)p Fx(-bimo)s(dules)e Fv(M)38 b Fx(and)28 b Fv(N)38 b Fx(arising)28 b(from)f Fn(K)p Fx(-mo)s(dules)h(as)g(ab)s(o) m(v)m(e,)i(i.e.)42 b(satisfying)0 4949 y Fv(\025m)28 b Fx(=)f Fv(m\025)p Fx(,)32 b(then)f(their)g(tensor)g(pro)s(duct)f Fv(M)f Fq(\012)1779 4964 y Fm(K)1853 4949 y Fv(N)41 b Fx(also)30 b(satis\014es)j(this)e(equation,)g(so)g Fv(M)e Fq(\012)3452 4964 y Fm(K)3526 4949 y Fv(N)41 b Fx(comes)0 5065 y(from)34 b(a)g(\(left\))h Fn(K)p Fx(-mo)s(dule.)49 b(Indeed)36 b(w)m(e)g(ha)m(v)m(e)f Fv(\025m)24 b Fq(\012)g Fv(n)31 b Fx(=)g Fv(m\025)23 b Fq(\012)h Fv(n)31 b Fx(=)g Fv(m)24 b Fq(\012)g Fv(\025n)30 b Fx(=)h Fv(m)24 b Fq(\012)g Fv(n\025)p Fx(.)49 b(Th)m(us)36 b(w)m(e)0 5182 y(can)d(also)f(de\014ne) i(a)f(tensor)g(pro)s(duct)g(of)f(t)m(w)m(o)h(left)g Fn(K)p Fx(-mo)s(dules.)0 5298 y(W)-8 b(e)35 b(often)g(write)h(the)g(tensor)f (pro)s(duct)g(of)g(t)m(w)m(o)h(v)m(ector)g(spaces)g(or)f(t)m(w)m(o)g (left)h(mo)s(dules)g Fv(M)45 b Fx(and)35 b Fv(N)46 b Fx(o)m(v)m(er)0 5414 y(a)33 b(comm)m(utativ)m(e)i(ring)e Fn(K)g Fx(as)h Fv(M)f Fq(\012)23 b Fv(N)43 b Fx(instead)34 b(of)f Fv(M)g Fq(\012)2098 5429 y Fm(K)2177 5414 y Fv(N)43 b Fx(and)34 b(the)f(tensor)h(pro)s(duct)f(o)m(v)m(er)h Fn(K)g Fx(of)e(t)m(w)m(o)0 5530 y Fn(K)p Fx(-mo)s(dule)h(homomorphisms) h Fv(f)44 b Fx(and)32 b Fv(g)k Fx(as)d Fv(f)g Fq(\012)22 b Fv(g)36 b Fx(instead)e(of)e Fv(f)h Fq(\012)2534 5545 y Fm(K)2612 5530 y Fv(g)t Fx(.)0 5646 y(\()p Fs(Warning:)43 b Fx(Do)32 b(not)g(confuse)i(this)f(with)g(a)f(tensor)i Fv(f)e Fq(\012)23 b Fv(g)t Fx(.)43 b(See)33 b(the)g(follo)m(wing)g (exercise.\))p eop end %%Page: 10 10 TeXDict begin 10 9 bop 0 -170 a Fu(10)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Ft(Problem)39 b(1.6.)j Fx(\(1\))34 b(Let)g Fv(M)1100 44 y Fr(R)1158 29 y Fx(,)1219 44 y Fr(R)1277 29 y Fv(N)10 b Fx(,)34 b Fv(M)1530 -7 y Fp(0)1520 55 y Fr(R)1578 29 y Fx(,)g(and)1830 44 y Fr(R)1888 29 y Fv(N)1976 -7 y Fp(0)2033 29 y Fx(b)s(e)g Fv(R)q Fx(-mo)s(dules.)48 b(Sho)m(w)34 b(that)g(the)g(follo)m(wing)g (is)g(a)0 146 y(homomorphism)g(of)e(Ab)s(elian)h(groups:)176 328 y Fv(\026)27 b Fx(:)h(Hom)520 343 y Fr(R)578 328 y Fx(\()p Fv(M)5 b(;)17 b(M)863 287 y Fp(0)887 328 y Fx(\))22 b Fq(\012)1024 343 y Fm(Z)1095 328 y Fx(Hom)1298 343 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b(dules)33 b(then)g(ther)-5 b(e)33 b(is)g(an)g(isomorphism)e(of)i Fn(K)p Fs(-mo)-5 b(dules)315 1546 y Fv(\034)39 b Fx(:)27 b Fv(M)33 b Fq(\012)23 b Fv(N)793 1518 y Fq(\030)794 1550 y Fx(=)898 1546 y Fv(N)33 b Fq(\012)23 b Fv(M)10 b Fs(.)148 1662 y Fx(\(5\))42 b(Existence)30 b(of)c(Inner)j(Hom-F)-8 b(unctors:)42 b Fs(L)-5 b(et)1945 1677 y Fr(R)2003 1662 y Fv(M)2097 1677 y Fr(T)2152 1662 y Fs(,)2213 1677 y Fr(S)2264 1662 y Fv(N)2342 1677 y Fr(T)2397 1662 y Fs(,)31 b(and)2642 1677 y Fr(S)2693 1662 y Fv(P)2756 1677 y Fr(R)2844 1662 y Fs(b)-5 b(e)30 b(bimo)-5 b(dules.)42 b(Then)29 b(ther)-5 b(e)315 1778 y(ar)g(e)34 b(c)-5 b(anonic)g(al)34 b(isomorphisms)f(of)h (bimo)-5 b(dules)635 1960 y Fx(Hom)838 1975 y Fr(S)t Fs(-)p Fr(T)975 1960 y Fx(\()p Fv(:P)35 b Fq(\012)1215 1975 y Fr(R)1296 1960 y Fv(M)5 b(:;)17 b(:N)5 b(:)p Fx(\))1669 1933 y Fq(\030)1670 1964 y Fx(=)1774 1960 y(Hom)1977 1975 y Fr(S)t Fs(-)p Fr(R)2116 1960 y Fx(\()p Fv(:P)s(:;)17 b(:)g Fx(Hom)2565 1975 y Fr(T)2620 1960 y Fx(\()p Fv(M)5 b(:;)17 b(N)5 b(:)p Fx(\))p Fv(:)p Fx(\))35 b Fs(and)731 2168 y Fx(Hom)934 2183 y Fr(S)t Fs(-)o Fr(T)1071 2168 y Fx(\()p Fv(:P)g Fq(\012)1311 2183 y Fr(R)1391 2168 y Fv(M)5 b(:;)17 b(:N)5 b(:)p Fx(\))1765 2140 y Fq(\030)1765 2172 y Fx(=)1870 2168 y(Hom)2073 2183 y Fr(R)p Fs(-)p Fr(T)2216 2168 y Fx(\()p Fv(:M)g(:;)17 b(:)g Fx(Hom)2698 2183 y Fr(S)2749 2168 y Fx(\()p Fv(:P)s(;)g(:N)10 b Fx(\))p Fv(:)p Fx(\))p Fv(:)0 2357 y Fs(Pr)-5 b(o)g(of.)41 b Fx(W)-8 b(e)33 b(only)g(describ)s(e)h(the)f(corresp)s(onding)h (homomorphisms.)0 2473 y(\(1\))e(Use)i(1.7)e(\(5\))g(to)g(de\014ne)i Fv(\013)q Fx(\(\()p Fv(m)22 b Fq(\012)h Fv(n)p Fx(\))f Fq(\012)h Fv(p)p Fx(\))k(:=)h Fv(m)22 b Fq(\012)h Fx(\()p Fv(n)f Fq(\012)h Fv(p)p Fx(\).)0 2589 y(\(2\))32 b(De\014ne)h Fv(\025)28 b Fx(:)g Fv(R)23 b Fq(\012)773 2604 y Fr(R)853 2589 y Fv(M)39 b Fq(\000)-60 b(!)27 b Fv(M)44 b Fx(b)m(y)33 b Fv(\025)p Fx(\()p Fv(r)25 b Fq(\012)d Fv(m)p Fx(\))28 b(:=)g Fv(r)s(m)p Fx(.)0 2705 y(\(3\))k(De\014ne)h Fv(\032)28 b Fx(:)g Fv(M)33 b Fq(\012)796 2720 y Fr(S)869 2705 y Fv(S)h Fq(\000)-60 b(!)28 b Fv(M)43 b Fx(b)m(y)33 b Fv(\032)p Fx(\()p Fv(m)23 b Fq(\012)g Fv(s)p Fx(\))k(:=)h Fv(ms)p Fx(.)0 2821 y(\(4\))k(De\014ne)h Fv(\034)11 b Fx(\()p Fv(m)23 b Fq(\012)g Fv(n)p Fx(\))28 b(:=)f Fv(n)c Fq(\012)f Fv(m)p Fx(.)0 2938 y(\(5\))33 b(F)-8 b(or)32 b Fv(f)40 b Fx(:)28 b Fv(P)36 b Fq(\012)652 2953 y Fr(R)733 2938 y Fv(M)k Fq(\000)-60 b(!)29 b Fv(N)43 b Fx(de\014ne)35 b Fv(\036)p Fx(\()p Fv(f)11 b Fx(\))28 b(:)h Fv(P)42 b Fq(\000)-60 b(!)28 b Fx(Hom)2148 2953 y Fr(T)2203 2938 y Fx(\()p Fv(M)5 b(;)17 b(N)10 b Fx(\))33 b(b)m(y)h Fv(\036)p Fx(\()p Fv(f)11 b Fx(\)\()p Fv(p)p Fx(\)\()p Fv(m)p Fx(\))28 b(:=)h Fv(f)11 b Fx(\()p Fv(p)22 b Fq(\012)h Fv(m)p Fx(\))34 b(and)0 3054 y Fv( )t Fx(\()p Fv(f)11 b Fx(\))27 b(:)h Fv(M)38 b Fq(\000)-59 b(!)27 b Fx(Hom)764 3069 y Fr(S)815 3054 y Fx(\()p Fv(P)s(;)17 b(N)10 b Fx(\))32 b(b)m(y)i Fv( )t Fx(\()p Fv(f)11 b Fx(\)\()p Fv(m)p Fx(\)\()p Fv(p)p Fx(\))27 b(:=)h Fv(f)11 b Fx(\()p Fv(p)22 b Fq(\012)g Fv(m)p Fx(\).)1503 b Fo(\003)0 3251 y Fx(Usually)36 b(one)f(iden)m (ti\014es)i(threefold)e(tensor)h(pro)s(ducts)f(along)g(the)g(map)g Fv(\013)g Fx(so)g(that)g(w)m(e)h(can)f(use)g Fv(M)g Fq(\012)3849 3266 y Fr(S)0 3367 y Fv(N)15 b Fq(\012)170 3382 y Fr(T)231 3367 y Fv(P)41 b Fx(:=)27 b(\()p Fv(M)15 b Fq(\012)689 3382 y Fr(S)746 3367 y Fv(N)10 b Fx(\))5 b Fq(\012)954 3382 y Fr(T)1014 3367 y Fv(P)41 b Fx(=)28 b Fv(M)15 b Fq(\012)1408 3382 y Fr(S)1464 3367 y Fx(\()p Fv(N)g Fq(\012)1672 3382 y Fr(T)1733 3367 y Fv(P)f Fx(\).)40 b(F)-8 b(or)23 b(the)i(notion)f(of)f(a)h(monoidal)g(or)g(tensor)h(category)-8 b(,)0 3483 y(ho)m(w)m(ev)m(er,)38 b(this)e(canonical)g(isomorphism)h (\(natural)e(transformation\))g(is)h(of)f(cen)m(tral)h(imp)s(ortance)g (and)0 3600 y(will)d(b)s(e)g(discussed)i(later.)0 3788 y Ft(Problem)j(1.7.)0 3905 y Fx(\(1\))j(Giv)m(e)h(a)f(complete)h(pro)s (of)e(of)h(Theorem)i(1.22.)68 b(In)42 b(\(5\))f(sho)m(w)h(ho)m(w)g(Hom) 2948 3920 y Fr(T)3003 3905 y Fx(\()p Fv(M)5 b(:;)17 b(N)5 b(:)p Fx(\))41 b(b)s(ecomes)i(an)0 4021 y Fv(S)6 b Fx(-)p Fv(R)q Fx(-bimo)s(dule.)0 4137 y(\(2\))32 b(Giv)m(e)h(an)g(explicit)h (pro)s(of)e(of)g Fv(M)h Fq(\012)1435 4152 y Fr(R)1515 4137 y Fx(\()p Fv(X)d Fq(\010)23 b Fv(Y)e Fx(\))1908 4109 y Fq(\030)1908 4141 y Fx(=)2013 4137 y Fv(M)33 b Fq(\012)2217 4152 y Fr(R)2297 4137 y Fv(X)d Fq(\010)22 b Fv(M)33 b Fq(\012)2711 4152 y Fr(R)2792 4137 y Fv(Y)21 b Fx(.)0 4253 y(\(3\))26 b(Sho)m(w)i(that)f(for)f(ev)m(ery)i(\014nite)g (dimensional)g(v)m(ector)g(space)g Fv(V)48 b Fx(there)28 b(is)f(a)f Fs(unique)h Fx(elemen)m(t)3497 4179 y Fl(P)3602 4205 y Fr(n)3602 4282 y(i)p Fk(=1)3737 4253 y Fv(v)3784 4268 y Fr(i)3823 4253 y Fq(\012)0 4370 y Fv(v)51 4333 y Fp(\003)47 4394 y Fr(i)118 4370 y Fq(2)h Fv(V)44 b Fq(\012)22 b Fv(V)491 4333 y Fp(\003)563 4370 y Fx(suc)m(h)34 b(that)e(the)h(follo)m(wing)g(holds)1376 4573 y Fq(8)p Fv(v)f Fq(2)c Fv(V)49 b Fx(:)1862 4478 y Fl(X)1922 4688 y Fr(i)2023 4573 y Fv(v)2074 4532 y 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4239 y Fv(V)5 b(;)17 b(:W)d Fx(\).)148 4376 y(\(5\))42 b Fn(Z)p Fv(=)p Fx(\(18\))21 b Fq(\012)702 4391 y Fm(Z)773 4376 y Fn(Z)p Fv(=)p Fx(\(30\))28 b Fq(6)p Fx(=)f(0.)148 4514 y(\(6\))42 b Fv(m)28 b Fx(:)f Fn(Z)p Fv(=)p Fx(\(18\))22 b Fq(\012)870 4529 y Fm(Z)941 4514 y Fn(Z)p Fv(=)p Fx(\(30\))27 b Fq(3)p 1352 4459 56 4 v 29 w Fv(x)22 b Fq(\012)p 1529 4459 52 4 v 23 w Fv(y)31 b Fq(7!)p 1735 4459 107 4 v 27 w Fv(xy)g Fq(2)d Fn(Z)p Fv(=)p Fx(\(6\))33 b(is)g(a)f(homomorphism)i (and)e Fv(m)h Fx(is)g(bijectiv)m(e.)148 4652 y(\(7\))42 b(F)-8 b(or)31 b Fn(Q)p Fx(-v)m(ector)j(spaces)g Fv(V)54 b Fx(and)33 b Fv(W)46 b Fx(w)m(e)34 b(ha)m(v)m(e)g Fv(V)43 b Fq(\012)2176 4667 y Fm(Z)2247 4652 y Fv(W)2381 4624 y Fq(\030)2381 4656 y Fx(=)2486 4652 y Fv(V)g Fq(\012)2663 4667 y Fm(Q)2742 4652 y Fv(W)14 b Fx(.)148 4789 y(\(8\))42 b(F)-8 b(or)31 b(eac)m(h)j(\014nite)f(Ab)s(elian)g(group)g Fv(M)43 b Fx(w)m(e)33 b(ha)m(v)m(e)h Fn(Q)23 b Fq(\012)2280 4804 y Fm(Z)2351 4789 y Fv(M)39 b Fx(=)27 b(0.)148 4927 y(\(9\))42 b Fn(Z)p Fv(=)p Fx(\()p Fv(m)p Fx(\))22 b Fq(\012)690 4942 y Fm(Z)761 4927 y Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))1038 4899 y Fq(\030)1039 4931 y Fx(=)1143 4927 y Fn(Z)p Fv(=)p Fx(\(ggT\()p Fv(m;)17 b(n)p Fx(\)\).)100 5065 y(\(10\))41 b Fn(Q)22 b Fq(\012)491 5080 y Fm(Z)562 5065 y Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))28 b(=)g(0.)100 5202 y(\(11\))41 b(Hom)518 5217 y Fm(Z)566 5202 y Fx(\()p Fn(Q)p Fv(;)17 b Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\)\))28 b(=)g(0.)100 5340 y(\(12\))41 b(Determine)33 b(explicitely)i (isomorphisms)g(for)1650 5529 y Fn(Z)22 b Fq(\012)1815 5544 y Fm(Z)1886 5529 y Fn(Q)1992 5501 y Fq(\030)1992 5533 y Fx(=)2097 5529 y Fn(Q)p Fv(;)1650 5645 y Fx(3)p Fn(Z)g Fq(\012)1864 5660 y Fm(Z)1935 5645 y Fn(Q)2040 5617 y Fq(\030)2041 5649 y Fx(=)2146 5645 y Fn(Q)p Fv(:)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 0 -170 a Fu(14)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)315 29 y Fx(Sho)m(w)33 b(that)f(the)h (follo)m(wing)g(diagram)g(comm)m(utes)1680 653 y Fn(Q)410 b(Q)p 1786 629 353 4 v 2056 627 a Fj(-)1924 710 y Fx(3)p Fq(\001)1537 166 y Fx(3)p Fn(Z)22 b Fq(\012)1751 181 y Fm(Z)1822 166 y Fn(Q)150 b(Z)22 b Fq(\012)2214 181 y Fm(Z)2285 166 y Fn(Q)p 1929 141 91 4 v 1937 139 a Fj(-)p 1717 559 4 351 v 1718 559 a(?)1602 386 y Fq(\030)1603 418 y Fx(=)p 2204 559 V 2206 559 a Fj(?)2245 386 y Fq(\030)2246 418 y Fx(=)100 833 y(\(13\))41 b(The)28 b(homomorphism)g(2)p Fn(Z)11 b Fq(\012)1396 848 y Fr(Z)1466 833 y Fn(Z)p Fv(=)p Fx(\(2\))27 b Fq(\000)-59 b(!)27 b Fn(Z)11 b Fq(\012)2032 848 y Fm(Z)2093 833 y Fn(Z)p Fv(=)p Fx(\(2\))27 b(is)h(the)g(zero)g (homomorphism,)i(but)d(b)s(oth)315 949 y(mo)s(dules)33 b(are)g(di\013eren)m(t)h(from)e(zero.)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 1545 -170 a Fu(Algebras)27 b(and)e(Coalgebras) 1445 b(15)1222 29 y Fx(2.)49 b Fw(Algebras)36 b(and)i(Co)n(algebras)0 204 y Fx(2.1.)49 b Ft(Algebras.)g Fx(Let)25 b Fn(K)h Fx(b)s(e)f(a)g(comm)m(utativ)m(e)i(ring.)41 b(W)-8 b(e)26 b(consider)g(all)f Fn(K)p Fx(-mo)s(dules)h(as)g Fn(K)p Fx(-)p Fn(K)p Fx(-bimo)s(dules)0 320 y(as)43 b(in)h(Remark)g(1.21.)75 b(T)-8 b(ensor)44 b(pro)s(ducts)g(of)f Fn(K)p Fx(-mo)s(dules)h(will)g (b)s(e)g(simply)h(written)f(as)f Fv(M)e Fq(\012)30 b Fv(N)56 b Fx(:=)0 436 y Fv(M)33 b Fq(\012)204 451 y Fr(K)295 436 y Fv(N)10 b Fx(.)0 624 y Ft(De\014nition)48 b(2.1.)e Fx(A)c Fn(K)p Fs(-algebr)-5 b(a)40 b Fx(is)i(a)f Fn(K)p Fx(-mo)s(dule)h Fv(A)f Fx(together)g(with)h(a)f Fs(multiplic)-5 b(ation)40 b Fq(r)j Fx(:)f Fv(A)28 b Fq(\012)h Fv(A)0 740 y Fq(\000)-60 b(!)28 b Fv(A)k Fx(\()p Fn(K)p Fx(-mo)s(dule)h (homomorphism\))h(that)f(is)g(asso)s(ciativ)m(e:)1475 1426 y Fv(A)22 b Fq(\012)h Fv(A)609 b(A)p 1772 1398 552 4 v 2241 1396 a Fj(-)2018 1461 y Fp(r)1377 939 y Fv(A)23 b Fq(\012)f Fv(A)g Fq(\012)h Fv(A)415 b(A)22 b Fq(\012)h Fv(A)p 1869 910 357 4 v 2143 908 a Fj(-)1955 883 y Fk(id)12 b Fp(\012r)p 1607 1328 4 351 v 1609 1328 a Fj(?)1397 1172 y Fp(r\012)p Fk(id)p 2387 1328 V 2389 1328 a Fj(?)2428 1175 y Fp(r)0 1607 y Fx(and)33 b(a)f Fs(unit)h Fv(\021)e Fx(:)d Fn(K)g Fq(\000)-59 b(!)27 b Fv(A)33 b Fx(\()p Fn(K)p Fx(-mo)s(dule)g(homomorphism\):)1175 1812 y Fn(K)22 b Fq(\012)h Fv(A)1475 1784 y Fq(\030)1476 1816 y Fx(=)1580 1812 y Fv(A)1681 1784 y Fq(\030)1682 1816 y Fx(=)1786 1812 y Fv(A)f Fq(\012)h Fn(K)400 b Fv(A)22 b Fq(\012)g Fv(A)p 2087 1783 342 4 v 2346 1781 a Fj(-)2176 1750 y Fk(id)12 b Fp(\012)p Fr(\021)p 1615 2201 4 351 v 1617 2201 a Fj(?)1426 2042 y Fr(\021)r Fp(\012)p Fk(id)p 2590 2201 V 2592 2201 a Fj(?)2631 2048 y Fp(r)1483 2299 y Fv(A)22 b Fq(\012)g Fv(A)791 b(A:)p 1779 2271 734 4 v 2430 2269 a Fj(-)2117 2334 y Fp(r)2166 2049 y Fk(id)1753 1892 y Fj(H)1836 1933 y(H)1919 1975 y(H)2002 2016 y(H)2085 2058 y(H)2168 2099 y(H)2251 2141 y(H)2334 2182 y(H)2372 2201 y(H)-83 b(j)0 2475 y Fx(A)33 b Fn(K)p Fx(-algebra)f Fv(A)h Fx(is)g Fs(c)-5 b(ommutative)32 b Fx(if)g(the)h(follo)m(wing)g (diagram)f(comm)m(utes)1572 2643 y Fv(A)23 b Fq(\012)f Fv(A)220 b(A)22 b Fq(\012)h Fv(A)p 1869 2614 162 4 v 1948 2612 a Fj(-)1931 2594 y Fr(\034)1900 3134 y Fv(A:)1710 2879 y Fp(r)1740 2764 y Fj(A)1782 2847 y(A)1823 2930 y(A)1865 3013 y(A)1874 3032 y(A)-42 b(U)2131 2879 y Fp(r)2118 2764 y Fj(\001)2077 2847 y(\001)2035 2930 y(\001)1994 3013 y(\001)1984 3032 y(\001)g(\013)0 3280 y Fx(Let)46 b Fv(A)g Fx(and)g Fv(B)51 b Fx(b)s(e)46 b Fn(K)p Fx(-algebras.)83 b(A)46 b Fs(homomorphism)f(of)i(algebr)-5 b(as)45 b Fv(f)61 b Fx(:)50 b Fv(A)g Fq(\000)-59 b(!)50 b Fv(B)g Fx(is)d(a)e Fn(K)p Fx(-mo)s(dule)0 3396 y(homomorphism)34 b(suc)m(h)g(that)e(the)h (follo)m(wing)g(diagrams)g(comm)m(ute:)1619 4087 y Fv(A)509 b(B)p 1721 4055 452 4 v 2090 4053 a Fj(-)1926 4118 y Fr(f)1522 3595 y Fv(A)22 b Fq(\012)h Fv(A)311 b(B)27 b Fq(\012)c Fv(B)p 1819 3567 254 4 v 1990 3565 a Fj(-)1877 3533 y Fr(f)7 b Fp(\012)p Fr(f)p 1654 3984 4 351 v 1656 3984 a Fj(?)1508 3826 y Fp(r)1567 3837 y Fi(A)p 2239 3984 V 2241 3984 a Fj(?)2280 3826 y Fp(r)2339 3837 y Fi(B)0 4272 y Fx(and)1903 4371 y Fn(K)1675 4589 y Fr(\021)1710 4600 y Fi(A)1866 4488 y Fj(\001)1825 4571 y(\001)1783 4654 y(\001)1742 4737 y(\001)1732 4756 y(\001)-42 b(\013)2123 4589 y Fr(\021)2158 4600 y Fi(B)1976 4488 y Fj(A)2017 4571 y(A)2059 4654 y(A)2100 4737 y(A)2110 4756 y(A)g(U)1661 4858 y Fv(A)398 b(B)5 b(:)p 1763 4826 341 4 v 2021 4824 a Fj(-)1912 4792 y Fr(f)0 5012 y Ft(Remark)38 b(2.2.)k Fx(Ev)m(ery)34 b Fn(K)p Fx(-algebra)f Fv(A)f Fx(is)h(a)g(ring)f(with)h (the)g(m)m(ultiplication)1459 5215 y Fv(A)23 b Fq(\002)f Fv(A)1786 5158 y Fp(\012)1755 5215 y Fq(\000)-59 b(!)27 b Fv(A)22 b Fq(\012)h Fv(A)2225 5158 y Fp(r)2196 5215 y Fq(\000)-60 b(!)27 b Fv(A:)0 5395 y Fx(The)34 b(unit)e(elemen)m(t)j (is)e Fv(\021)t Fx(\(1\),)f(where)i(1)e(is)h(the)g(unit)g(elemen)m(t)h (of)f Fn(K)p Fx(.)0 5511 y(Ob)m(viously)i(the)f(comp)s(osition)g(of)e (t)m(w)m(o)i(homomorphisms)h(of)e(algebras)h(is)f(again)g(a)g (homomorphism)h(of)0 5627 y(algebras.)44 b(F)-8 b(urthermore)33 b(the)g(iden)m(tit)m(y)h(map)f(is)g(a)f(homomorphism)i(of)e(algebras.)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 0 -170 a Fu(16)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Ft(Problem)38 b(2.1.)k Fx(\(1\))32 b(Sho)m(w)h(that)f(End)1470 44 y Fm(K)1526 29 y Fx(\()p Fv(V)22 b Fx(\))32 b(is)h(a)f Fn(K)p Fx(-algebra.)0 146 y(\(2\))39 b(Sho)m(w)h(that)g(\()p Fv(A;)17 b Fq(r)39 b Fx(:)g Fv(A)27 b Fq(\012)g Fv(A)40 b Fq(\000)-60 b(!)39 b Fv(A;)17 b(\021)43 b Fx(:)d Fn(K)g Fq(\000)-60 b(!)39 b Fv(A)p Fx(\))g(is)h(a)g Fn(K)p Fx(-algebra)f(if)g(and)h(only)g(if)f Fv(A)g Fx(with)i(the)0 291 y(m)m(ultiplication)35 b Fv(A)23 b Fq(\002)g Fv(A)957 234 y Fp(\012)926 291 y Fq(\000)-60 b(!)29 b Fv(A)23 b Fq(\012)h Fv(A)1400 234 y Fp(r)1371 291 y Fq(\000)-59 b(!)29 b Fv(A)34 b Fx(and)f(the)h(unit)g Fv(\021)t Fx(\(1\))f(is)h(a)g(ring)f(and)h Fv(\021)f Fx(:)d Fn(K)g Fq(\000)-60 b(!)29 b Fx(Cen)m(t)r(\()p Fv(A)p Fx(\))k(is)h(a)0 407 y(ring)f(homomorphism)g(in)m(to)g(the)g Fs(c)-5 b(enter)32 b Fx(of)h Fv(A)p Fx(,)f(where)i(Cen)m(t)q(\()p Fv(A)p Fx(\))28 b(:=)g Fq(f)p Fv(a)f Fq(2)h 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b(!)54 b Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))3588 4539 y Fr(op)3710 4575 y Fx(with)0 4691 y Fv(S)6 b Fx(\()p Fv(v)t Fx(\))36 b(=)h Fq(\000)p Fv(v)t Fx(.)61 b(\()p Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))821 4655 y Fr(op)933 4691 y Fx(is)38 b(the)h Fs(opp)-5 b(osite)39 b(algebr)-5 b(a)37 b Fx(of)h Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))38 b(with)h(m)m(ultiplication)g Fv(s)26 b Fq(\003)g Fv(t)37 b Fx(:=)g Fv(ts)h Fx(for)g(all)0 4807 y Fv(s;)17 b(t)28 b Fq(2)g Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))27 b(=)h Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))828 4771 y Fr(op)934 4807 y Fx(and)33 b(where)h Fv(st)f Fx(denotes)g(the)g (pro)s(duct)g(in)g Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\).\))0 4924 y(\(7\))32 b(Sho)m(w)i(that)e(the)h(diagrams)1447 5092 y Fv(T)14 b Fx(\()p Fv(V)22 b Fx(\))238 b Fn(K)p 1701 5069 182 4 v 1800 5067 a Fj(-)1776 5048 y Fr(")2227 5092 y Fv(T)14 b Fx(\()p Fv(V)22 b Fx(\))p 2017 5069 V 2116 5067 a Fj(-)2089 5035 y Fr(\021)1274 5482 y Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))h Fq(\012)h Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))208 b Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))h Fq(\012)h Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))p 1875 5459 151 4 v 1943 5457 a Fj(-)1882 5431 y Fk(1)p Fp(\012)p Fr(S)1882 5521 y(S)t Fp(\012)p Fk(1)p 1558 5389 4 254 v 1560 5389 a Fj(?)1462 5285 y Fk(\001)p 2338 5389 V 2340 5218 a Fj(6)2379 5285 y Fp(r)0 5646 y Fx(comm)m(ute.)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 1545 -170 a Fu(Algebras)27 b(and)e(Coalgebras) 1445 b(19)0 29 y Fx(2.3.)49 b Ft(Symmetric)38 b(algebras.)0 248 y(De\014nition)43 b(2.8.)g Fx(Let)37 b Fn(K)g Fx(b)s(e)f(a)h(comm)m (utativ)m(e)h(ring.)55 b(Let)37 b Fv(V)58 b Fx(b)s(e)36 b(a)g Fn(K)p Fx(-mo)s(dule.)56 b(A)37 b Fn(K)p Fx(-algebra)f Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))0 365 y(together)k(with)h(a)f (homomorphism)h(of)f Fn(K)p Fx(-mo)s(dules)h Fv(\023)i Fx(:)g Fv(V)49 b Fq(\000)-59 b(!)27 b Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\),)27 b(suc)m(h)g(that)e Fv(\023)p Fx(\()p Fv(v)t Fx(\))7 b Fq(\001)g Fv(\023)p Fx(\()p Fv(v)3321 329 y Fp(0)3344 365 y Fx(\))27 b(=)h Fv(\023)p Fx(\()p Fv(v)3636 329 y Fp(0)3659 365 y Fx(\))7 b Fq(\001)g Fv(\023)p Fx(\()p Fv(v)t Fx(\))0 481 y(for)35 b(all)h Fv(v)t(;)17 b(v)437 445 y Fp(0)492 481 y Fq(2)33 b Fv(V)22 b Fx(,)36 b(is)g(called)h(a)e Fs(symmetric)i(algebr)-5 b(a)37 b(over)e Fv(V)57 b Fx(if)36 b(for)f(eac)m(h)h Fn(K)p Fx(-algebra)g Fv(A)f Fx(and)h(for)f(eac)m(h)0 597 y(homomorphism)26 b(of)f Fn(K)p Fx(-mo)s(dules)h Fv(f)39 b Fx(:)28 b Fv(V)49 b Fq(\000)-60 b(!)28 b Fv(A)p Fx(,)e(suc)m(h)h(that)e Fv(f)11 b Fx(\()p Fv(v)t Fx(\))c Fq(\001)g Fv(f)k Fx(\()p Fv(v)2582 561 y Fp(0)2604 597 y Fx(\))28 b(=)f Fv(f)11 b Fx(\()p Fv(v)2921 561 y Fp(0)2944 597 y Fx(\))c Fq(\001)g Fv(f)k Fx(\()p Fv(v)t Fx(\))24 b(for)h(all)g Fv(v)t(;)17 b(v)3650 561 y Fp(0)3700 597 y Fq(2)28 b Fv(V)22 b Fx(,)0 713 y(there)33 b(exists)i(a)d(unique)i(homomorphism)g(of)e Fn(K)p Fx(-algebras)h Fv(g)e Fx(:)d Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))27 b Fq(\000)-59 b(!)27 b Fv(A)33 b Fx(suc)m(h)h(that)e (the)h(diagram)1632 951 y Fv(V)360 b(S)6 b Fx(\()p Fv(V)21 b Fx(\))p 1739 928 281 4 v 1937 926 a Fj(-)1863 907 y Fv(\023)1796 1194 y(f)1739 1077 y Fj(@)1822 1160 y(@)1905 1243 y(@)1988 1326 y(@)2007 1345 y(@)-83 b(R)2122 1447 y Fv(A)p 2157 1345 4 351 v 2159 1345 a Fj(?)2198 1181 y Fv(g)0 1648 y Fx(comm)m(utes.)0 1765 y Fs(Note:)53 b Fx(If)37 b(y)m(ou)h(w)m(an)m(t)g(to)f(de\014ne)i(a)e(homomorphism)h Fv(g)h Fx(:)d Fv(S)6 b Fx(\()p Fv(V)22 b Fx(\))35 b Fq(\000)-59 b(!)35 b Fv(A)i Fx(with)h(a)f(symmetric)j(algebra)d(as)0 1881 y(domain)31 b(y)m(ou)h(should)g(de\014ne)g(it)f(b)m(y)h(giving)f (a)g(homomorphism)h(of)f Fn(K)p Fx(-mo)s(dules)h Fv(f)38 b Fx(:)28 b Fv(V)49 b Fq(\000)-59 b(!)27 b Fv(A)k Fx(satisfying)0 1997 y Fv(f)11 b Fx(\()p Fv(v)t Fx(\))21 b Fq(\001)h Fv(f)11 b Fx(\()p Fv(v)405 1961 y Fp(0)428 1997 y Fx(\))28 b(=)f Fv(f)11 b Fx(\()p Fv(v)745 1961 y Fp(0)768 1997 y Fx(\))22 b Fq(\001)g Fv(f)11 b Fx(\()p Fv(v)t Fx(\))32 b(for)g(all)g Fv(v)t(;)17 b(v)1526 1961 y Fp(0)1576 1997 y Fq(2)28 b Fv(V)22 b Fx(.)0 2216 y Ft(Lemma)39 b(2.9.)j Fs(A)35 b(symmetric)g(algebr)-5 b(a)34 b Fx(\()p Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(\023)p Fx(\))35 b Fs(de\014ne)-5 b(d)34 b(by)h Fv(V)57 b Fs(is)34 b(unique)h(up)g(to)g(a) g(unique)g(isomor-)0 2332 y(phism.)0 2551 y(Pr)-5 b(o)g(of.)41 b Fx(Let)33 b(\()p Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(\023)p Fx(\))33 b(and)f(\()p Fv(S)1173 2515 y Fp(0)1196 2551 y Fx(\()p Fv(V)22 b Fx(\))p Fv(;)17 b(\023)1429 2515 y Fp(0)1452 2551 y Fx(\))33 b(b)s(e)g(symmetric)h(algebras)f(o)m (v)m(er)h Fv(V)21 b Fx(.)44 b(Then)1905 2770 y Fv(V)1360 3002 y(\023)1725 2846 y Fj(\010)1642 2888 y(\010)1559 2929 y(\010)1476 2971 y(\010)1393 3012 y(\010)1310 3054 y(\010)1227 3095 y(\010)1143 3137 y(\010)1106 3156 y(\010)-83 b(\031)1761 3017 y Fv(\023)1795 2981 y Fp(0)1793 2888 y Fj(\000)1710 2971 y(\000)1627 3054 y(\000)1544 3137 y(\000)1525 3156 y(\000)g(\011)2093 3002 y Fv(\023)2012 2888 y Fj(@)2095 2971 y(@)2178 3054 y(@)2261 3137 y(@)2280 3156 y(@)g(R)2494 3017 y Fv(\023)2528 2981 y Fp(0)2081 2846 y Fj(H)2164 2888 y(H)2247 2929 y(H)2330 2971 y(H)2413 3012 y(H)2496 3054 y(H)2579 3095 y(H)2662 3137 y(H)2700 3156 y(H)g(j)859 3249 y Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))256 b Fv(S)1401 3213 y Fp(0)1424 3249 y Fx(\()p Fv(V)21 b Fx(\))p 1108 3226 199 4 v 1224 3224 a Fj(-)1179 3205 y Fv(h)p 1607 3226 687 4 v 2211 3224 a Fj(-)1923 3205 y Fv(k)2322 3249 y(S)6 b Fx(\()p Fv(V)21 b Fx(\))255 b Fv(S)2863 3213 y Fp(0)2886 3249 y Fx(\()p Fv(V)22 b Fx(\))p 2570 3226 199 4 v 2686 3224 a Fj(-)2641 3205 y Fv(h)0 3480 y Fx(implies)34 b Fv(k)d Fx(=)c Fv(h)572 3444 y Fp(\000)p Fk(1)667 3480 y Fx(.)3129 b Fo(\003)0 3699 y Ft(Prop)s(osition)54 b(2.10.)49 b Fx(\(Rules)f(of)e(computation) h(in)g(a)g(symmetric)i(algebra\))f Fs(L)-5 b(et)48 b Fx(\()p Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(\023)p Fx(\))48 b Fs(b)-5 b(e)48 b(the)0 3815 y(symmetric)34 b(algebr)-5 b(a)34 b(over)h Fv(V)21 b Fs(.)45 b(Then)34 b(we)h(have)148 3991 y Fx(\(1\))42 b Fv(\023)28 b Fx(:)f Fv(V)50 b Fq(\000)-57 b(!)27 b Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))35 b Fs(is)g(inje)-5 b(ctive)34 b(\(we)g(wil)5 b(l)34 b(identify)h(the)g(elements)f Fv(\023)p Fx(\()p Fv(v)t Fx(\))h Fs(and)f Fv(v)39 b Fs(for)34 b(al)5 b(l)35 b Fv(v)c Fq(2)d Fv(V)22 b Fs(\),)148 4109 y Fx(\(2\))42 b Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))27 b(=)h Fq(f)716 4034 y Fl(P)821 4138 y Fr(n;)p 884 4085 25 3 v(i)928 4109 y Fv(v)975 4124 y Fr(i)999 4133 y Fg(1)1060 4109 y Fq(\001)22 b Fv(:)17 b(:)g(:)22 b Fq(\001)g Fv(v)1344 4124 y Fr(i)1368 4132 y Fi(n)1414 4109 y Fq(j)p 1442 4032 34 4 v Fv(i)28 b Fx(=)g(\()p Fv(i)1678 4124 y Fk(1)1717 4109 y Fv(;)17 b(:)g(:)g(:)f(;)h(i)1969 4124 y Fr(n)2016 4109 y Fx(\))35 b Fs(multiindex)f(of)g(length)h Fv(n)p Fq(g)p Fv(;)148 4234 y Fx(\(3\))42 b Fs(if)35 b Fv(f)40 b Fx(:)30 b Fv(V)51 b Fq(\000)-57 b(!)29 b Fv(A)36 b Fs(is)f(a)h(homomorphism)d(of)j Fn(K)p Fs(-mo)-5 b(dules)35 b(satisfying)h Fv(f)11 b Fx(\()p Fv(v)t Fx(\))22 b Fq(\001)g Fv(f)11 b Fx(\()p Fv(v)3237 4198 y Fp(0)3260 4234 y Fx(\))29 b(=)h Fv(f)11 b Fx(\()p Fv(v)3581 4198 y Fp(0)3603 4234 y Fx(\))23 b Fq(\001)g Fv(f)11 b Fx(\()p Fv(v)t Fx(\))315 4350 y Fs(for)34 b(al)5 b(l)34 b Fv(v)t(;)17 b(v)755 4314 y Fp(0)805 4350 y Fq(2)28 b Fv(V)22 b Fs(,)34 b Fv(A)h Fs(is)f(a)g Fn(K)p Fs(-algebr)-5 b(a,)34 b(and)g Fv(g)d Fx(:)d Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))28 b Fq(\000)-57 b(!)27 b Fv(A)35 b Fs(is)f(the)g(induc)-5 b(e)g(d)34 b(homomorphism)315 4466 y Fn(K)p Fs(-algebr)-5 b(as,)34 b(then)1026 4718 y Fv(g)t Fx(\()1115 4623 y Fl(X)1143 4848 y Fr(n;)p 1206 4795 25 3 v(i)1275 4718 y Fv(v)1322 4733 y Fr(i)1346 4742 y Fg(1)1407 4718 y Fq(\001)22 b Fv(:)17 b(:)g(:)22 b Fq(\001)g Fv(v)1691 4733 y Fr(i)1715 4741 y Fi(n)1761 4718 y Fx(\))28 b(=)1931 4623 y Fl(X)1959 4848 y Fr(n;)p 2022 4795 V(i)2091 4718 y Fv(f)11 b Fx(\()p Fv(v)2235 4733 y Fr(i)2259 4742 y Fg(1)2298 4718 y Fx(\))22 b Fq(\001)g Fv(:)17 b(:)g(:)k Fq(\001)h Fv(f)11 b Fx(\()p Fv(v)2738 4733 y Fr(i)2762 4741 y Fi(n)2808 4718 y Fx(\))p Fv(:)0 5057 y Fs(Pr)-5 b(o)g(of.)41 b Fx(\(1\))31 b(Use)h(the)g(em)m(b) s(edding)g(homomorphism)h Fv(j)g Fx(:)28 b Fv(V)49 b Fq(\000)-59 b(!)27 b Fv(D)s Fx(\()p Fv(V)21 b Fx(\),)32 b(where)g Fv(D)s Fx(\()p Fv(V)21 b Fx(\))31 b(is)h(the)g(comm)m(uta-)0 5173 y(tiv)m(e)i(algebra)e(de\014ned)j(in)e(2.1)f(\(3\))g(to)h (construct)g Fv(g)f Fx(:)c Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))28 b Fq(\000)-60 b(!)28 b Fv(D)s Fx(\()p Fv(V)21 b Fx(\))33 b(suc)m(h)h(that)e Fv(g)26 b Fq(\016)c Fv(\023)28 b Fx(=)g Fv(j)6 b Fx(.)44 b(Since)34 b Fv(j)k Fx(is)0 5289 y(injectiv)m(e)c(so)f(is)g Fv(\023)p Fx(.)0 5405 y(\(2\))h(Let)g Fv(B)i Fx(:=)31 b Fq(f)629 5330 y Fl(P)734 5434 y Fr(n;)p 797 5381 V(i)841 5405 y Fv(v)888 5420 y Fr(i)912 5429 y Fg(1)974 5405 y Fq(\001)23 b Fv(:)17 b(:)g(:)23 b Fq(\001)g Fv(v)1261 5420 y Fr(i)1285 5428 y Fi(n)1332 5405 y Fq(j)p 1360 5328 34 4 v Fv(i)31 b Fx(=)f(\()p Fv(i)1601 5420 y Fk(1)1641 5405 y Fv(;)17 b(:)g(:)g(:)f(;)h(i)1893 5420 y Fr(n)1940 5405 y Fx(\))32 b(m)m(ultiindex)j(of)d(length)h Fv(n)p Fq(g)p Fx(.)49 b(Ob)m(viously)36 b Fv(B)k Fx(is)35 b(the)0 5530 y(subalgebra)i(of)e Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))36 b(generated)h(b)m(y)g(the)f(elemen) m(ts)i(of)d Fv(V)22 b Fx(.)53 b(Let)36 b Fv(j)k Fx(:)33 b Fv(B)39 b Fq(\000)-60 b(!)33 b Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))36 b(b)s(e)g(the)g(em)m(b)s(edding)0 5646 y(homomorphism.)70 b(Then)42 b Fv(\023)g Fx(:)g Fv(V)64 b Fq(\000)-60 b(!)42 b Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))41 b(factors)g(through)g(a)f Fn(K)p Fx(-mo)s(dule)i(homomorphism)g Fv(\023)3687 5610 y Fp(0)3752 5646 y Fx(:)g Fv(V)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 0 -170 a Fu(20)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fq(\000)-60 b(!)28 b Fv(B)5 b Fx(.)43 b(The)34 b(follo)m(wing)f(diagram)1376 252 y Fv(V)430 b(B)p 1483 220 351 4 v 1751 218 a Fj(-)1630 199 y Fv(\023)1664 163 y Fp(0)2280 243 y Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))p 1971 220 281 4 v 2169 218 a Fj(-)2088 180 y Fv(j)1863 731 y(B)343 b(S)6 b Fx(\()p Fv(V)21 b Fx(\))p 1971 708 V 2169 706 a Fj(-)2088 667 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5627 y(Pr)g(o)g(of.)41 b Fx(Use)34 b(that)e Fn(K)h Fx(itself)g(is)g(a)g Fn(K)p Fx(-algebra.)2154 b Fo(\003)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 0 -170 a Fu(26)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Ft(Remark)39 b(2.24.)j Fx(If)33 b(w)m(e)i(write)f(the)f(ev)-5 b(aluation)34 b(as)g Fv(C)2027 -7 y Fp(\003)2089 29 y Fq(\012)23 b Fv(C)36 b Fq(3)29 b Fv(a)23 b Fq(\012)g Fv(c)29 b Fq(7!)f(h)p Fv(a;)17 b(c)p Fq(i)29 b(2)g Fn(K)34 b Fx(then)g(an)f(elemen)m(t)0 146 y Fv(a)i Fq(2)g Fv(C)264 110 y Fp(\003)340 146 y Fx(is)i(completely)h(determined)h(b)m(y)f(the)f(v)-5 b(alues)37 b(of)f Fq(h)p Fv(a;)17 b(c)p Fq(i)36 b Fx(for)g(all)h Fv(c)e Fq(2)g Fv(C)7 b Fx(.)55 b(So)37 b(the)g(pro)s(duct)g(of)f Fv(a)0 262 y Fx(and)d Fv(b)g Fx(in)f Fv(C)454 226 y Fp(\003)526 262 y Fx(is)h(uniquely)i(determined)g(b)m(y)e(the)g(form)m(ula)1024 438 y Fq(h)p Fv(a)22 b Fq(\003)g Fv(b;)17 b(c)p Fq(i)28 b Fx(=)f Fq(h)p Fv(a)22 b Fq(\012)h Fv(b;)17 b Fx(\001\()p Fv(c)p Fx(\))p Fq(i)28 b Fx(=)2171 343 y Fl(X)2332 438 y Fv(a)p Fx(\()p Fv(c)2463 454 y Fk(\(1\))2557 438 y Fx(\))p Fv(b)p Fx(\()p Fv(c)2716 454 y Fk(\(2\))2811 438 y Fx(\))p Fv(:)0 614 y Fx(The)34 b(unit)e(elemen)m(t)j(of)d Fv(C)955 578 y Fp(\003)1027 614 y Fx(is)h Fv(\017)28 b Fq(2)g Fv(C)1363 578 y Fp(\003)1402 614 y Fx(.)0 791 y Ft(Lemma)58 b(2.25.)51 b Fs(L)-5 b(et)51 b Fn(K)f Fs(b)-5 b(e)50 b(a)h(\014eld)e(and)h Fv(A)h Fs(b)-5 b(e)50 b(a)g(\014nite)g (dimensional)e Fn(K)p Fs(-algebr)-5 b(a.)91 b(Then)49 b Fv(A)3728 754 y Fp(\003)3824 791 y Fx(=)0 907 y(Hom)203 922 y Fm(K)259 907 y Fx(\()p Fv(A;)17 b Fn(K)p Fx(\))35 b Fs(is)g(a)f Fn(K)p Fs(-c)-5 b(o)g(algebr)g(a.)0 1083 y(Pr)g(o)g(of.)41 b Fx(De\014ne)33 b(the)g(com)m(ultiplication)h(on)f Fv(A)1695 1047 y Fp(\003)1767 1083 y Fx(b)m(y)1203 1277 y(\001)28 b(:)g Fv(A)1440 1235 y Fp(\003)1540 1220 y(r)1599 1197 y Fh(\003)1507 1277 y Fq(\000)-16 b(!)28 b Fx(\()p Fv(A)22 b Fq(\012)g Fv(A)p Fx(\))2039 1235 y Fp(\003)2107 1220 y Fk(can)2212 1197 y Fh(\000)p Fg(1)2120 1277 y Fq(\000)-16 b(!)42 b Fv(A)2396 1235 y Fp(\003)2457 1277 y Fq(\012)23 b Fv(A)2630 1235 y Fp(\003)2670 1277 y Fv(:)0 1437 y Fx(The)31 b(canonical)f(map)g(can)e(:)g Fv(A)1139 1401 y Fp(\003)1194 1437 y Fq(\012)16 b Fv(A)1360 1401 y Fp(\003)1428 1437 y Fq(\000)-59 b(!)27 b Fx(\()p Fv(A)16 b Fq(\012)g Fv(A)p Fx(\))1904 1401 y Fp(\003)1974 1437 y Fx(is)30 b(in)m(v)m(ertible,)j(since)e Fv(A)f Fx(is)g(\014nite)g (dimensional.)44 b(By)0 1553 y(a)33 b(diagrammatic)h(pro)s(of)e(or)h(b) m(y)i(calculation)f(with)f(elemen)m(ts)j(it)e(is)f(easy)i(to)e(sho)m(w) h(that)g Fv(A)3390 1517 y Fp(\003)3462 1553 y Fx(b)s(ecomes)h(a)0 1669 y Fn(K)p Fx(-coalgebra.)3285 b Fo(\003)0 1845 y Ft(Remark)50 b(2.26.)d Fx(If)c Fn(K)g Fx(is)h(an)f(arbitrary)g(comm)m (utativ)m(e)i(ring)e(and)g Fv(A)g Fx(is)g(a)g Fn(K)p Fx(-algebra,)j(then)d Fv(A)3739 1809 y Fp(\003)3824 1845 y Fx(=)0 1962 y(Hom)203 1977 y Fm(K)259 1962 y Fx(\()p Fv(A;)17 b Fn(K)p Fx(\))33 b(is)g(a)f Fn(K)p Fx(-coalgebra)h(if)f Fv(A)h Fx(is)g(a)f(\014nitely)i(generated)f(pro)5 b(jectiv)m(e)35 b Fn(K)p Fx(-mo)s(dule.)0 2138 y Ft(Problem)47 b(2.8.)f Fx(Find)40 b(su\016cien)m(t)j(conditions)e(for)f(an)g(algebra)h Fv(A)f Fx(resp.)68 b(a)40 b(coalgebra)h Fv(C)47 b Fx(suc)m(h)42 b(that)0 2254 y(Hom\()p Fv(A;)17 b(C)7 b Fx(\))32 b(b)s(ecomes)i(a)f (coalgebra)f(with)h(co-con)m(v)m(olution)h(as)f(com)m(ultiplication.)0 2430 y(2.7.)49 b Ft(Como)s(dules.)0 2607 y(De\014nition)c(2.27.)h Fx(Let)39 b Fv(C)46 b Fx(b)s(e)39 b(a)f Fn(K)p Fx(-coalgebra.)63 b(A)39 b Fs(left)i Fv(C)7 b Fs(-c)-5 b(omo)g(dule)37 b Fx(is)j(a)f Fn(K)p Fx(-mo)s(dule)g Fv(M)50 b Fx(together)0 2723 y(with)33 b(a)f Fn(K)p Fx(-mo)s(dule)i(homomorphism)f Fv(\016)1489 2738 y Fr(M)1596 2723 y Fx(:)28 b Fv(M)39 b Fq(\000)-60 b(!)27 b Fv(C)i Fq(\012)23 b Fv(M)10 b Fx(,)33 b(suc)m(h)h(that)f(the)g(diagrams)1359 3376 y Fv(C)c Fq(\012)22 b Fv(M)388 b(C)29 b Fq(\012)23 b Fv(C)29 b Fq(\012)23 b Fv(M)p 1691 3348 320 4 v 1928 3346 a Fj(-)1771 3412 y Fk(id)12 b Fp(\012)p Fr(\016)1458 2889 y Fv(M)587 b(C)29 b Fq(\012)22 b Fv(M)p 1591 2860 519 4 v 2027 2858 a Fj(-)1834 2840 y Fr(\016)p 1509 3278 4 351 v 1510 3278 a Fj(?)1438 3126 y Fr(\016)p 2289 3278 V 2290 3278 a Fj(?)2329 3122 y Fk(\001)p Fp(\012)p Fk(id)0 3540 y Fx(and)1345 3619 y Fv(M)p 1396 4004 V 1398 4004 a Fj(?)1325 3851 y Fr(\016)1246 4102 y Fv(C)29 b Fq(\012)23 b Fv(M)552 b Fn(K)23 b Fq(\012)f Fv(M)2423 4074 y Fq(\030)2423 4106 y Fx(=)2528 4102 y Fv(M)5 b(:)p 1578 4074 485 4 v 1980 4072 a Fj(-)1749 4137 y Fr(\017)p Fp(\012)p Fk(id)1948 3851 y(id)1534 3694 y Fj(H)1617 3736 y(H)1700 3777 y(H)1783 3819 y(H)1866 3860 y(H)1949 3902 y(H)2032 3943 y(H)2115 3985 y(H)2153 4004 y(H)-83 b(j)0 4237 y Fx(comm)m(ute.)0 4354 y(Let)175 4317 y Fr(C)234 4354 y Fv(M)44 b Fx(and)562 4317 y Fr(C)621 4354 y Fv(N)g Fx(b)s(e)33 b Fv(C)7 b Fx(-como)s(dules)34 b(and)f(let)g Fv(f)39 b Fx(:)29 b Fv(M)39 b Fq(\000)-60 b(!)28 b Fv(N)44 b Fx(b)s(e)33 b(a)f Fn(K)p Fx(-mo)s(dule)i(homomorphism.)46 b(The)0 4470 y(map)33 b Fv(f)43 b Fx(is)33 b(called)g(a)g Fs(homomorphism)f(of) j(c)-5 b(omo)g(dules)31 b Fx(if)i(the)g(diagram)1471 5135 y Fv(N)603 b(C)29 b Fq(\012)22 b Fv(N)p 1588 5107 535 4 v 2040 5105 a Fj(-)1811 5170 y Fr(\016)1842 5181 y Fi(N)1463 4647 y Fv(M)586 b(C)29 b Fq(\012)23 b Fv(M)p 1596 4619 519 4 v 2032 4617 a Fj(-)1806 4587 y Fr(\016)1837 4598 y Fi(M)p 1514 5037 4 351 v 1515 5037 a Fj(?)1435 4878 y Fr(f)p 2293 5037 V 2295 5037 a Fj(?)2334 4878 y Fk(1)p Fp(\012)p Fr(f)0 5294 y Fx(comm)m(utes.)0 5411 y(Let)k Fv(N)38 b Fx(b)s(e)27 b(an)g(arbitrary)g Fn(K)p Fx(-mo)s(dule)h(and)f Fv(M)38 b Fx(b)s(e)28 b(a)e Fv(C)7 b Fx(-como)s(dule.)42 b(Then)29 b(there)e(is)h(a)f(bijection)h(b)s(et)m (w)m(een)0 5527 y(all)k(m)m(ultilinear)i(maps)1380 5646 y Fv(f)k Fx(:)28 b Fv(C)h Fq(\002)23 b Fv(:)17 b(:)g(:)k Fq(\002)i Fv(C)29 b Fq(\002)23 b Fv(M)38 b Fq(\000)-60 b(!)28 b Fv(N)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 1545 -170 a Fu(Algebras)27 b(and)e(Coalgebras) 1445 b(27)0 29 y Fx(and)33 b(all)f(linear)h(maps)1357 169 y Fv(f)1416 128 y Fp(0)1467 169 y Fx(:)28 b Fv(C)h Fq(\012)22 b Fv(:)17 b(:)g(:)22 b Fq(\012)h Fv(C)29 b Fq(\012)22 b Fv(M)39 b Fq(\000)-60 b(!)28 b Fv(N)5 b(:)0 330 y Fx(These)34 b(maps)g(are)e(asso)s(ciated)i(to)e(eac)m(h)h(other)g (b)m(y)h(the)f(form)m(ula)1098 510 y Fv(f)11 b Fx(\()p Fv(c)1237 525 y Fk(1)1276 510 y Fv(;)17 b(:)g(:)g(:)f(;)h(c)1537 525 y Fr(n)1583 510 y Fv(;)g(m)p Fx(\))28 b(=)g Fv(f)1941 469 y Fp(0)1964 510 y Fx(\()p Fv(c)2044 525 y Fk(1)2105 510 y Fq(\012)23 b Fv(:)17 b(:)g(:)k Fq(\012)i Fv(c)2483 525 y Fr(n)2552 510 y Fq(\012)g Fv(m)p Fx(\))p Fv(:)0 691 y Fx(F)-8 b(or)32 b Fv(m)c Fq(2)g Fv(M)43 b Fx(w)m(e)34 b(de\014ne)1079 778 y Fl(X)1240 872 y Fv(f)11 b Fx(\()p Fv(m)1422 888 y Fk(\(1\))1516 872 y Fv(;)17 b(:)g(:)g(:)f(;)h(m)1820 888 y Fk(\()p Fr(n)p Fk(\))1922 872 y Fv(;)g(m)2051 888 y Fk(\()p Fr(M)7 b Fk(\))2185 872 y Fx(\))27 b(:=)h Fv(f)2440 831 y Fp(0)2463 872 y Fx(\()p Fv(\016)2548 831 y Fr(n)2595 872 y Fx(\()p Fv(m)p 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1806 y Fk(\(1\))1179 1791 y Fq(\012)g Fv(m)1364 1806 y Fk(\(2\))1481 1791 y Fq(\012)f Fv(m)1665 1806 y Fk(\()p Fr(M)7 b Fk(\))1827 1791 y Fx(=)28 b(\(1)22 b Fq(\012)g Fv(\016)t Fx(\))p Fv(\016)t Fx(\()p Fv(c)p Fx(\))28 b(=)f(\(\001)22 b Fq(\012)h Fx(1\))p Fv(\016)t Fx(\()p Fv(m)p Fx(\))p Fv(:)0 1995 y Ft(Problem)33 b(2.9.)39 b Fx(Sho)m(w)30 b(that)f(a)f(\014nite)i (dimensional)g(v)m(ector)g(space)g Fv(V)51 b Fx(is)29 b(a)g(como)s(dule)g(o)m(v)m(er)h(the)f(coalge-)0 2111 y(bra)24 b Fv(V)i Fq(\012)t Fv(V)408 2075 y Fp(\003)471 2111 y Fx(as)e(de\014ned)i(in)e(exercise)i(2.7)d(\(1\))h(with)g(the)h (coaction)f Fv(\016)t Fx(\()p Fv(v)t Fx(\))j(:=)2745 2036 y Fl(P)2866 2111 y Fv(v)8 b Fq(\012)t Fv(v)3053 2075 y Fp(\003)3049 2136 y Fr(i)3098 2111 y Fq(\012)t Fv(v)3226 2126 y Fr(i)3283 2111 y Fq(2)28 b Fx(\()p Fv(V)e Fq(\012)t Fv(V)3658 2075 y Fp(\003)3697 2111 y Fx(\))t Fq(\012)t Fv(V)0 2227 y Fx(where)282 2153 y Fl(P)404 2227 y Fv(v)455 2191 y Fp(\003)451 2252 y Fr(i)516 2227 y Fq(\012)d Fv(v)663 2242 y Fr(i)723 2227 y Fx(is)33 b(the)g(dual)g(basis)h(of)e Fv(V)54 b Fx(in)33 b Fv(V)1860 2191 y Fp(\003)1922 2227 y Fq(\012)22 b Fv(V)g Fx(.)0 2415 y Ft(Theorem)43 b(2.28.)g Fs(\(F)-7 b(undamental)37 b(The)-5 b(or)g(em)37 b(for)i(Como)-5 b(dules\))37 b(L)-5 b(et)38 b Fn(K)h Fs(b)-5 b(e)39 b(a)f(\014eld.)55 b(L)-5 b(et)38 b Fv(M)50 b Fs(b)-5 b(e)38 b(a)g(left)0 2531 y Fv(C)7 b Fs(-c)-5 b(omo)g(dule)41 b(and)g(let)h Fv(m)g Fq(2)f Fv(M)53 b Fs(b)-5 b(e)41 b(given.)66 b(Then)41 b(ther)-5 b(e)42 b(exists)g(a)f(\014nite)h(dimensional)e(sub)-5 b(c)g(o)g(algebr)g(a)0 2648 y Fv(C)77 2612 y Fp(0)136 2648 y Fq(\022)36 b Fv(C)47 b Fs(and)38 b(a)h(\014nite)g(dimensional)f Fv(C)1529 2612 y Fp(0)1552 2648 y Fs(-c)-5 b(omo)g(dule)38 b Fv(M)2116 2612 y Fp(0)2180 2648 y Fs(with)h Fv(m)d Fq(2)g Fv(M)2723 2612 y Fp(0)2783 2648 y Fq(\022)g Fv(M)50 b Fs(wher)-5 b(e)39 b Fv(M)3424 2612 y Fp(0)3484 2648 y Fq(\022)d Fv(M)50 b Fs(is)39 b(a)0 2764 y Fn(K)p Fs(-submo)-5 b(dule,)34 b(such)h(that)g(the)g(diagr)-5 b(am)1452 3444 y Fv(M)587 b(C)29 b Fq(\012)22 b Fv(M)p 1585 3416 519 4 v 2021 3414 a Fj(-)1828 3479 y Fr(\016)1440 2959 y Fv(M)1544 2922 y Fp(0)2109 2959 y Fv(C)2186 2922 y Fp(0)2232 2959 y Fq(\012)g Fv(M)2435 2922 y Fp(0)p 1597 2928 484 4 v 1998 2926 a Fj(-)1811 2907 y Fr(\016)1844 2883 y Fh(0)p 1503 3345 4 351 v 1504 3345 a Fj(?)p 2283 3345 V 697 w(?)0 3612 y Fs(c)-5 b(ommutes.)0 3800 y Ft(Corollary)36 b(2.29.)k Fs(\(1\))32 b(Each)g(element)f Fv(c)d Fq(2)g Fv(C)39 b Fs(of)32 b(a)g(c)-5 b(o)g(algebr)g(a)31 b(is)i(c)-5 b(ontaine)g(d)31 b(in)h(a)g(\014nite)g(dimensional)0 3916 y(sub)-5 b(c)g(o)g(algebr)g(a)34 b(of)g Fv(C)7 b Fs(.)0 4032 y(\(2\))34 b(Each)f(element)h Fv(m)28 b Fq(2)g Fv(M)45 b Fs(of)34 b(a)g(c)-5 b(omo)g(dule)33 b(is)h(c)-5 b(ontaine)g(d)33 b(in)h(a)g(\014nite)g(dimensional)e(sub)-5 b(c)g(omo)g(dule)33 b(of)0 4149 y Fv(M)10 b Fs(.)0 4337 y Ft(Corollary)38 b(2.30.)j Fs(\(1\))33 b(Each)h(\014nite)f (dimensional)f(subsp)-5 b(ac)g(e)33 b Fv(V)56 b Fs(of)33 b(a)h(c)-5 b(o)g(algebr)g(a)33 b Fv(C)40 b Fs(is)34 b(c)-5 b(ontaine)g(d)33 b(in)g(a)0 4453 y(\014nite)i(dimensional)e(sub)-5 b(c)g(o)g(algebr)g(a)33 b Fv(C)1418 4417 y Fp(0)1476 4453 y Fs(of)i Fv(C)7 b Fs(.)0 4569 y(\(2\))30 b(Each)f(\014nite)h (dimensional)e(subsp)-5 b(ac)g(e)29 b Fv(V)52 b Fs(of)29 b(a)h(c)-5 b(omo)g(dule)29 b Fv(M)41 b Fs(is)30 b(c)-5 b(ontaine)g(d)29 b(in)g(a)h(\014nite)g(dimensional)0 4685 y(sub)-5 b(c)g(omo)g(dule)34 b Fv(M)657 4649 y Fp(0)716 4685 y Fs(of)g Fv(M)10 b Fs(.)0 4873 y Ft(Corollary)39 b(2.31.)j Fs(\(1\))35 b(Each)f(c)-5 b(o)g(algebr)g(a)34 b(is)g(a)h(union)g(of)f(\014nite)h(dimensional)e(sub)-5 b(c)g(o)g(algebr)g(as.)0 4989 y(\(2\))34 b(Each)h(c)-5 b(omo)g(dule)34 b(is)g(a)h(union)f(of)h(\014nite)g(dimensional)e(sub)-5 b(c)g(omo)g(dules.)0 5177 y(Pr)g(o)g(of.)41 b Fx(\()p Fs(of)34 b(the)h(The)-5 b(or)g(em)p Fx(\))30 b(W)-8 b(e)33 b(can)f(assume)h(that)f Fv(m)c Fq(6)p Fx(=)g(0)k(for)f(else)i(w)m(e)g (can)g(use)g Fv(M)3168 5141 y Fp(0)3219 5177 y Fx(=)28 b(0)k(and)g Fv(C)3670 5141 y Fp(0)3721 5177 y Fx(=)27 b(0.)0 5294 y(Under)j(the)h(represen)m(tations)h(of)d Fv(\016)t Fx(\()p Fv(m)p Fx(\))f Fq(2)g Fv(C)23 b Fq(\012)16 b Fv(M)41 b Fx(as)30 b(\014nite)g(sums)h(of)f(decomp)s(osable)h (tensors)g(pic)m(k)g(one)1547 5554 y Fv(\016)t Fx(\()p Fv(m)p Fx(\))d(=)1942 5430 y Fr(s)1887 5460 y Fl(X)1902 5670 y Fr(i)p Fk(=1)2047 5554 y Fv(c)2089 5569 y Fr(i)2140 5554 y Fq(\012)22 b Fv(m)2324 5569 y Fr(i)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 0 -170 a Fu(28)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fx(of)39 b(shortest)i(length)g Fv(s)p Fx(.)65 b(Then)41 b(the)g(families)g(\()p Fv(c)1820 44 y Fr(i)1848 29 y Fq(j)p Fv(i)f Fx(=)g(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(s)p Fx(\))39 b(and)h(\()p Fv(m)2776 44 y Fr(i)2805 29 y Fq(j)p Fv(i)g Fx(=)g(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(s)p Fx(\))39 b(are)h(linearly)0 146 y(indep)s(enden)m(t.)46 b(Cho)s(ose)33 b(co)s(e\016cien)m(ts)i Fv(c)1466 161 y Fr(ij)1554 146 y Fq(2)28 b Fv(C)39 b Fx(suc)m(h)34 b(that)1182 436 y(\001\()p Fv(c)1343 451 y Fr(j)1380 436 y Fx(\))27 b(=)1608 311 y Fr(t)1549 341 y Fl(X)1564 551 y Fr(i)p Fk(=1)1709 436 y Fv(c)1751 451 y Fr(i)1802 436 y Fq(\012)22 b Fv(c)1943 451 y Fr(ij)2004 436 y Fv(;)114 b Fq(8)p Fv(j)34 b Fx(=)28 b(1)p Fv(;)17 b(:)g(:)g(:)e(;)i(s;)0 722 y Fx(b)m(y)31 b(suitably)h(extending)g(the)f (linearly)h(indep)s(enden)m(t)g(family)f(\()p Fv(c)2388 737 y Fr(i)2416 722 y Fq(j)p Fv(i)d Fx(=)g(1)p Fv(;)17 b(:)g(:)g(:)e(;)i(s)p Fx(\))30 b(to)g(a)h(linearly)g(indep)s(en-)0 838 y(den)m(t)i(family)h(\()p Fv(c)598 853 y Fr(i)626 838 y Fq(j)p Fv(i)27 b Fx(=)h(1)p Fv(;)17 b(:)g(:)g(:)e(;)i(t)p Fx(\))33 b(and)f Fv(t)c Fq(\025)h Fv(s)p Fx(.)0 955 y(W)-8 b(e)44 b(\014rst)f(sho)m(w)i(that)e(w)m(e)h(can)g(c)m(ho)s(ose)g Fv(t)i Fx(=)f Fv(s)p Fx(.)76 b(By)44 b(coasso)s(ciativit)m(y)h(w)m(e)f (ha)m(v)m(e)3095 880 y Fl(P)3201 906 y Fr(s)3201 984 y(i)p Fk(=1)3336 955 y Fv(c)3378 970 y Fr(i)3435 955 y Fq(\012)30 b Fv(\016)t Fx(\()p 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b(0)27 b(=)370 1697 y Fl(P)475 1723 y Fr(s)475 1801 y(j)t Fk(=1)618 1772 y Fv(c)660 1787 y Fr(ij)743 1772 y Fq(\012)22 b Fv(m)927 1787 y Fr(j)997 1772 y Fx(for)32 b Fv(i)c(>)f(s)p Fx(.)44 b(The)33 b(last)g(statemen)m(t)h(implies)1325 1970 y Fv(c)1367 1985 y Fr(ij)1455 1970 y Fx(=)27 b(0)p Fv(;)114 b Fq(8)p Fv(i)29 b(>)e(s;)17 b(j)34 b Fx(=)27 b(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(s:)0 2164 y Fx(Hence)34 b(w)m(e)g(get)e Fv(t)c Fx(=)g Fv(s)k Fx(and)1182 2433 y(\001\()p Fv(c)1343 2448 y Fr(j)1380 2433 y Fx(\))27 b(=)1605 2309 y Fr(s)1549 2339 y Fl(X)1564 2549 y Fr(i)p Fk(=1)1709 2433 y Fv(c)1751 2448 y Fr(i)1802 2433 y Fq(\012)22 b Fv(c)1943 2448 y Fr(ij)2004 2433 y Fv(;)114 b Fq(8)p Fv(j)34 b Fx(=)28 b(1)p Fv(;)17 b(:)g(:)g(:)e(;)i(s:)0 2720 y Fx(De\014ne)50 b(\014nite)f(dimensional)i(subspaces)g Fv(C)1677 2684 y Fp(0)1756 2720 y Fx(=)56 b Fq(h)p Fv(c)1969 2735 y Fr(ij)2029 2720 y Fq(j)p Fv(i;)17 b(j)61 b Fx(=)56 b(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(s)p Fq(i)55 b(\022)h Fv(C)g Fx(and)49 b Fv(M)3344 2684 y Fp(0)3424 2720 y Fx(=)55 b Fq(h)p Fv(m)3679 2735 y Fr(i)3707 2720 y Fq(j)p Fv(i)h Fx(=)0 2836 y(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(s)p Fq(i)49 b(\022)h Fv(M)10 b Fx(.)83 b(Then)47 b(b)m(y)g(\(3\))e(w)m(e)h (get)g Fv(\016)54 b Fx(:)c Fv(M)1940 2800 y Fp(0)2014 2836 y Fq(\000)-60 b(!)49 b Fv(C)2257 2800 y Fp(0)2312 2836 y Fq(\012)31 b Fv(M)2524 2800 y Fp(0)2548 2836 y Fx(.)83 b(W)-8 b(e)46 b(sho)m(w)g(that)g Fv(m)k Fq(2)g Fv(M)3673 2800 y Fp(0)3743 2836 y Fx(and)0 2952 y(that)32 b(the)g(restriction)i(of)d(\001)i(to)e Fv(C)1264 2916 y Fp(0)1320 2952 y Fx(giv)m(es)i(a)f Fn(K)p Fx(-mo)s(dule)h (homomorphism)g(\001)28 b(:)g Fv(C)3022 2916 y Fp(0)3073 2952 y Fq(\000)-60 b(!)27 b Fv(C)3294 2916 y Fp(0)3371 2952 y Fq(\012)22 b Fv(C)3547 2916 y Fp(0)3602 2952 y Fx(so)32 b(that)0 3069 y(the)e(required)h(prop)s(erties)f(of)f(the)h (theorem)g(are)g(satis\014ed.)44 b(First)29 b(observ)m(e)i(that)f Fv(m)e Fx(=)3223 2994 y Fl(P)3345 3069 y Fv(")p Fx(\()p Fv(c)3471 3084 y Fr(i)3499 3069 y Fx(\))p Fv(m)3622 3084 y Fr(i)3678 3069 y Fq(2)g Fv(M)3876 3032 y Fp(0)0 3185 y Fx(and)33 b Fv(c)232 3200 y Fr(j)296 3185 y Fx(=)399 3110 y Fl(P)521 3185 y Fv(")p Fx(\()p Fv(c)647 3200 y Fr(i)675 3185 y Fx(\))p Fv(c)755 3200 y Fr(ij)843 3185 y Fq(2)28 b Fv(C)1014 3149 y Fp(0)1037 3185 y Fx(.)44 b(Using)33 b(coasso)s(ciativit)m(y)h(w)m(e)g(get)824 3306 y Fl(P)929 3332 y Fr(s)929 3410 y(i;j)t Fk(=1)1116 3381 y Fv(c)1158 3396 y Fr(i)1208 3381 y Fq(\012)23 b Fx(\001\()p Fv(c)1469 3396 y Fr(ij)1529 3381 y Fx(\))g Fq(\012)f Fv(m)1774 3396 y Fr(j)1844 3381 y Fx(=)1947 3306 y Fl(P)2052 3332 y Fr(s)2052 3410 y(k)r(;j)t Fk(=1)2254 3381 y Fx(\001\()p Fv(c)2415 3396 y Fr(k)2457 3381 y Fx(\))h Fq(\012)f Fv(c)2659 3396 y Fr(k)r(j)2756 3381 y Fq(\012)h Fv(m)2941 3396 y Fr(j)1844 3504 y Fx(=)1947 3429 y Fl(P)2052 3456 y Fr(s)2052 3533 y(i;j;k)r Fk(=1)2293 3504 y Fv(c)2335 3519 y Fr(i)2386 3504 y Fq(\012)f Fv(c)2527 3519 y Fr(ik)2616 3504 y Fq(\012)h Fv(c)2758 3519 y Fr(k)r(j)2855 3504 y Fq(\012)f Fv(m)3039 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Fq(\012)14 b Fv(C)21 b Fq(\012)14 b Fv(M)0 4717 y Fq(\000)-60 b(!)43 b Fv(M)10 b Fx(.)71 b(Consider)43 b(the)f(submo)s(dule)h Fv(N)54 b Fx(:=)43 b Fv(C)1810 4680 y Fp(\003)1849 4717 y Fv(m)p Fx(.)71 b(Then)43 b Fv(N)52 b Fx(is)42 b(\014nite)g(dimensional,)j(since)e Fv(c)3656 4680 y Fp(\003)3696 4717 y Fv(m)g Fx(=)0 4758 y Fl(P)105 4784 y Fr(n)105 4862 y(i)p Fk(=1)224 4833 y Fq(h)p Fv(c)305 4797 y Fp(\003)344 4833 y Fv(;)17 b(c)430 4848 y Fr(i)458 4833 y Fq(i)p Fv(m)582 4848 y Fr(i)636 4833 y Fx(for)25 b(all)h Fv(c)949 4797 y Fp(\003)1016 4833 y Fq(2)i Fv(C)1187 4797 y Fp(\003)1253 4833 y Fx(where)1528 4758 y Fl(P)1633 4784 y Fr(n)1633 4862 y(i)p Fk(=1)1768 4833 y Fv(c)1810 4848 y Fr(i)1847 4833 y Fq(\012)9 b Fv(m)2018 4848 y Fr(i)2074 4833 y Fx(=)28 b Fv(\016)t Fx(\()p Fv(m)p Fx(\).)41 b(Observ)m(e)28 b(that)e Fv(C)3104 4797 y Fp(\003)3143 4833 y Fv(m)g Fx(is)h(a)e(subspace)j(of)0 4949 y(the)g(space)h(generated)g(b)m(y)g(the)f Fv(m)1238 4964 y Fr(i)1267 4949 y Fx(.)42 b(But)28 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(homomorphism)g Fv(\036)29 b Fx(:)g Fv(D)k Fq(\000)-60 b(!)29 b Fv(C)41 b Fx(b)m(y)34 b Fv(n)24 b Fq(\012)f Fv(n)2118 5610 y Fp(\003)2187 5646 y Fq(7!)2316 5572 y Fl(P)2438 5646 y Fv(n)2496 5662 y Fk(\(1\))2591 5646 y Fq(h)p Fv(n)2688 5610 y Fp(\003)2727 5646 y Fv(;)17 b(n)2829 5662 y Fk(\()p Fr(N)7 b Fk(\))2951 5646 y Fq(i)p Fx(.)46 b(By)35 b(de\014nition)f(of)g(the)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 1545 -170 a Fu(Algebras)27 b(and)e(Coalgebras) 1445 b(29)0 29 y Fx(dual)33 b(basis)g(w)m(e)h(ha)m(v)m(e)g Fv(n)28 b Fx(=)1014 -45 y Fl(P)1136 29 y Fv(n)1194 44 y Fr(i)1222 29 y Fq(h)p Fv(n)1319 -7 y Fp(\003)1319 54 y Fr(i)1359 29 y Fv(;)17 b(n)p Fq(i)p Fx(.)43 b(Th)m(us)34 b(w)m(e)g(get)492 185 y(\()p Fv(\036)21 b Fq(\012)i Fv(\036)p Fx(\)\001)886 200 y Fr(D)950 185 y Fx(\()p Fv(n)f Fq(\012)h Fv(n)1226 149 y Fp(\003)1266 185 y Fx(\))32 b(=)c(\()p Fv(\036)21 b Fq(\012)i Fv(\036)p Fx(\)\()1791 110 y Fl(P)1912 185 y Fv(n)g Fq(\012)f Fv(n)2150 149 y Fp(\003)2150 209 y Fr(i)2212 185 y Fq(\012)h Fv(n)2370 200 y Fr(i)2420 185 y Fq(\012)g Fv(n)2578 149 y Fp(\003)2618 185 y Fx(\))1336 301 y(=)1440 226 y Fl(P)1561 301 y Fv(n)1619 316 y Fk(\(1\))1714 301 y Fq(h)p Fv(n)1811 265 y Fp(\003)1811 326 y Fr(i)1850 301 y Fv(;)17 b(n)1952 316 y Fk(\()p Fr(N)7 b Fk(\))2074 301 y Fq(i)22 b(\012)h Fv(n)2293 316 y Fr(i)p Fk(\(1\))2411 301 y Fq(h)p Fv(n)2508 265 y Fp(\003)2548 301 y Fv(;)17 b(n)2650 316 y Fr(i)p Fk(\()p Fr(N)7 b Fk(\))2796 301 y Fq(i)1336 417 y Fx(=)1440 342 y Fl(P)1561 417 y Fv(n)1619 433 y Fk(\(1\))1736 417 y Fq(\012)23 b Fv(n)1894 433 y Fr(i)p Fk(\(1\))2012 417 y Fq(h)p Fv(n)2109 381 y Fp(\003)2149 417 y Fv(;)17 b(n)2251 433 y Fr(i)p Fk(\()p Fr(N)7 b Fk(\))2397 417 y Fq(ih)p Fv(n)2533 381 y Fp(\003)2533 442 y Fr(i)2572 417 y Fv(;)17 b(n)2674 433 y Fk(\()p Fr(N)7 b Fk(\))2796 417 y Fq(i)1336 533 y Fx(=)1440 459 y Fl(P)1561 533 y Fv(n)1619 549 y Fk(\(1\))1736 533 y Fq(\012)23 b Fv(n)1894 549 y Fk(\(2\))1988 533 y Fq(h)p Fv(n)2085 497 y Fp(\003)2125 533 y Fv(;)17 b(n)2227 549 y Fk(\()p Fr(N)7 b Fk(\))2349 533 y Fq(i)27 b Fx(=)2519 459 y Fl(P)2640 533 y Fx(\001)2721 548 y Fr(C)2781 533 y Fx(\()p Fv(n)2877 549 y Fk(\(1\))2971 533 y Fx(\))p Fq(h)p Fv(n)3106 497 y Fp(\003)3146 533 y Fv(;)17 b(n)3248 549 y Fk(\()p Fr(N)7 b Fk(\))3370 533 y Fq(i)1336 650 y Fx(=)28 b(\001)1521 665 y Fr(C)1580 650 y Fv(\036)p Fx(\()p Fv(n)22 b Fq(\012)h Fv(n)1914 613 y Fp(\003)1953 650 y Fx(\))p Fv(:)0 808 y Fx(F)-8 b(urthermore)33 b Fv(")611 823 y Fr(C)670 808 y Fv(\036)p Fx(\()p Fv(n)22 b Fq(\012)h Fv(n)1004 772 y Fp(\003)1043 808 y Fx(\))28 b(=)f Fv(")p Fx(\()1296 733 y Fl(P)1418 808 y Fv(n)1476 824 y Fk(\(1\))1570 808 y Fq(h)p Fv(n)1667 772 y Fp(\003)1707 808 y Fv(;)17 b(n)1809 824 y Fk(\()p Fr(N)7 b Fk(\))1931 808 y Fq(i)p Fx(\))27 b(=)h Fq(h)p Fv(n)2236 772 y Fp(\003)2275 808 y Fv(;)2319 733 y Fl(P)2441 808 y Fv(")p Fx(\()p Fv(n)2583 824 y Fk(\(1\))2677 808 y Fx(\))p Fv(n)2773 824 y Fk(\()p Fr(N)7 b Fk(\))2895 808 y Fq(i)28 b Fx(=)f Fq(h)p Fv(n)3162 772 y Fp(\003)3202 808 y Fv(;)17 b(n)p Fq(i)27 b Fx(=)h Fv(")p Fx(\()p Fv(n)22 b Fq(\012)g Fv(n)3795 772 y Fp(\003)3835 808 y Fx(\).)0 924 y(Hence)35 b Fv(\036)29 b Fx(:)g Fv(D)j Fq(\000)-60 b(!)29 b Fv(C)41 b Fx(is)34 b(a)f(homomorphism)i(of)e(coalgebras,)h Fv(D)i Fx(is)e(\014nite)g (dimensional)h(and)f(the)g(image)0 1041 y Fv(C)77 1005 y Fp(0)128 1041 y Fx(:=)28 b Fv(\036)p Fx(\()p Fv(D)s Fx(\))f(is)i(a)g(\014nite)g(dimensional)h(sub)s(coalgebra)f(of)f Fv(C)7 b Fx(.)42 b(Clearly)30 b Fv(N)39 b Fx(is)29 b(also)f(a)h Fv(C)3178 1005 y Fp(0)3201 1041 y Fx(-como)s(dule,)h(since)0 1157 y(it)j(is)g(a)f Fv(D)s Fx(-como)s(dule.)0 1273 y(Finally)i(w)m(e)h (sho)m(w)g(that)e(the)i Fv(D)s Fx(-como)s(dule)e(structure)i(on)f Fv(N)44 b Fx(if)34 b(lifted)g(to)g(the)g Fv(C)7 b Fx(-como)s(dule)34 b(structure)0 1389 y(coincides)g(with)g(the)f(one)f(de\014ned)i(on)f Fv(M)10 b Fx(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)276 1520 y Fv(\016)319 1535 y Fr(C)379 1520 y Fx(\()p Fv(c)459 1484 y Fp(\003)498 1520 y Fv(m)p Fx(\))g(=)27 b Fv(\016)800 1535 y Fr(C)860 1520 y Fx(\()898 1445 y Fl(P)1003 1520 y Fq(h)p Fv(c)1084 1484 y Fp(\003)1123 1520 y Fv(;)17 b(m)1252 1535 y Fk(\(1\))1346 1520 y Fq(i)p Fv(m)1470 1535 y Fk(\()p Fr(M)7 b Fk(\))1604 1520 y Fx(\))28 b(=)1773 1445 y Fl(P)1879 1520 y Fq(h)p Fv(c)1960 1484 y Fp(\003)1999 1520 y Fv(;)17 b(m)2128 1535 y Fk(\(1\))2222 1520 y Fq(i)p Fv(m)2346 1535 y Fk(\(2\))2463 1520 y Fq(\012)22 b Fv(m)2647 1535 y Fk(\()p Fr(M)7 b Fk(\))654 1636 y Fx(=)757 1561 y Fl(P)863 1636 y Fq(h)p Fv(c)944 1600 y Fp(\003)983 1636 y Fv(;)17 b(m)1112 1651 y Fk(\(1\))1206 1636 y Fq(i)p Fv(m)1330 1651 y Fk(\(2\))1447 1636 y Fq(\012)22 b Fv(m)1631 1651 y Fr(i)1660 1636 y Fq(h)p Fv(m)1784 1600 y Fp(\003)1784 1661 y Fr(i)1823 1636 y Fv(;)17 b(m)1952 1651 y Fk(\()p Fr(M)7 b Fk(\))2086 1636 y Fq(i)28 b Fx(=)2256 1561 y Fl(P)2361 1636 y Fq(h)p Fv(c)2442 1600 y Fp(\003)2481 1636 y Fv(;)17 b(m)2610 1651 y Fk(\(1\))2705 1636 y Fq(i)p Fv(m)2829 1651 y Fk(\(2\))2923 1636 y Fq(h)p Fv(m)3047 1600 y Fp(\003)3047 1661 y Fr(i)3087 1636 y Fv(;)g(m)3216 1651 y Fk(\()p Fr(M)7 b Fk(\))3350 1636 y Fq(i)21 b(\012)i Fv(m)3595 1651 y Fr(i)654 1752 y Fx(=)k(\()p Fv(\036)22 b Fq(\012)h Fx(1\)\()1100 1677 y Fl(P)1205 1752 y Fq(h)p Fv(c)1286 1716 y Fp(\003)1325 1752 y Fv(;)17 b(m)1454 1768 y Fk(\(1\))1548 1752 y Fq(i)p Fv(m)1672 1768 y Fk(\()p Fr(M)7 b Fk(\))1828 1752 y Fq(\012)23 b Fv(m)2013 1716 y Fp(\003)2013 1777 y Fr(i)2075 1752 y Fq(\012)f Fv(m)2259 1767 y Fr(i)2288 1752 y Fx(\))28 b(=)f(\()p Fv(\036)22 b Fq(\012)h Fx(1\)\()2800 1677 y Fl(P)2921 1752 y Fv(c)2963 1716 y Fp(\003)3002 1752 y Fv(m)g Fq(\012)f Fv(m)3294 1716 y Fp(\003)3294 1777 y Fr(i)3356 1752 y Fq(\012)h Fv(m)3541 1767 y Fr(i)3569 1752 y Fx(\))654 1868 y(=)k(\()p Fv(\036)22 b Fq(\012)h Fx(1\))p Fv(\016)1105 1883 y Fr(D)1169 1868 y Fx(\()p Fv(c)1249 1832 y Fp(\003)1288 1868 y Fv(m)p Fx(\))p Fv(:)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 0 -170 a Fu(30)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)942 29 y Fx(3.)49 b Fw(Pr)n(ojective)37 b(Modules)g(and)h(Genera)-7 b(tors)0 204 y Fx(3.1.)49 b Ft(Pro)s(ducts)37 b(and)h(copro)s(ducts.)0 392 y(De\014nition)g(3.1.) 0 508 y Fx(\(1\))g(Let)h(\()p Fv(M)476 523 y Fr(i)505 508 y Fq(j)p Fv(i)f Fq(2)h Fv(I)8 b Fx(\))38 b(b)s(e)h(a)g(family)g(of) g Fv(R)q Fx(-mo)s(dules.)62 b(An)40 b Fv(R)q Fx(-mo)s(dule)2655 433 y Fl(Q)2766 508 y Fv(M)2860 523 y Fr(i)2927 508 y Fx(together)f(with)h(a)e(family)0 624 y(of)e(homomorphisms)j(\()p Fv(p)934 639 y Fr(j)1006 624 y Fx(:)1068 549 y Fl(Q)1179 624 y Fv(M)1273 639 y Fr(i)1336 624 y Fq(\000)-59 b(!)35 b Fv(M)1583 639 y Fr(j)1620 624 y Fq(j)p Fv(j)40 b Fq(2)c Fv(I)8 b Fx(\))37 b(is)g(called)h(a)f Fs(\(dir)-5 b(e)g(ct\))38 b(pr)-5 b(o)g(duct)37 b Fx(of)g(the)g Fv(M)3505 639 y Fr(i)3570 624 y Fx(and)h(the)0 740 y(homomorphisms)k Fv(p)784 755 y Fr(j)862 740 y Fx(:)930 666 y Fl(Q)1041 740 y Fv(M)1135 755 y Fr(i)1205 740 y Fq(\000)-60 b(!)41 b Fv(M)1457 755 y Fr(j)1534 740 y Fx(are)g(called)g Fs(pr)-5 b(oje)g(ctions)p Fx(,)42 b(if)e(for)g(eac)m(h)h Fv(R)q Fx(-mo)s(dule)g Fv(N)51 b Fx(and)40 b(for)0 857 y(eac)m(h)30 b(family)g(of)f(homomorphisms)h(\()p Fv(f)1428 872 y Fr(j)1493 857 y Fx(:)d Fv(N)39 b Fq(\000)-60 b(!)27 b Fv(M)1902 872 y Fr(j)1939 857 y Fq(j)p Fv(j)34 b Fq(2)28 b Fv(I)8 b Fx(\))29 b(there)h(is)f(a)g(unique)i(homomorphism)f Fv(f)39 b Fx(:)27 b Fv(N)0 973 y Fq(\000)-60 b(!)145 898 y Fl(Q)255 973 y Fv(M)349 988 y Fr(i)410 973 y Fx(suc)m(h)35 b(that)1615 1499 y Fl(Q)1726 1574 y Fv(M)1820 1589 y Fr(i)2154 1574 y Fv(M)2248 1589 y Fr(j)p 1877 1552 248 4 v 2041 1550 a Fj(-)1958 1612 y Fv(p)2007 1627 y Fr(j)2036 1327 y Fv(f)2084 1342 y Fr(j)1800 1214 y Fj(@)1883 1297 y(@)1966 1380 y(@)2049 1463 y(@)2068 1482 y(@)-83 b(R)1687 1097 y Fv(N)p 1730 1482 4 351 v 1732 1482 a Fj(?)1634 1331 y Fv(f)0 1761 y Fx(comm)m(ute)34 b(for)e(all)h Fv(j)g Fq(2)28 b Fv(I)8 b Fx(.)0 1877 y(\(2\))39 b(\\The)h(dual)f(notion)g(is) h(called)g(copro)s(duct":)57 b(Let)40 b(\()p Fv(M)2213 1892 y Fr(i)2241 1877 y Fq(j)p Fv(i)f Fq(2)h Fv(I)8 b Fx(\))39 b(b)s(e)g(a)g(family)h(of)f Fv(R)q Fx(-mo)s(dules.)64 b(An)0 1994 y Fv(R)q Fx(-mo)s(dule)465 1919 y Fl(`)575 1994 y Fv(M)669 2009 y Fr(i)743 1994 y Fx(together)46 b(with)g(a)f(family)h(of)f(homomorphisms)i(\()p Fv(\023)2717 2009 y Fr(j)2804 1994 y Fx(:)j Fv(M)2975 2009 y Fr(j)3061 1994 y Fq(\000)-60 b(!)3228 1919 y Fl(`)3338 1994 y Fv(M)3432 2009 y Fr(i)3461 1994 y Fq(j)p Fv(j)55 b Fq(2)50 b Fv(I)8 b Fx(\))45 b(is)0 2110 y(called)c(a)f Fs(c)-5 b(opr)g(o)g(duct)39 b Fx(or)h Fs(dir)-5 b(e)g(ct)42 b(sum)d Fx(of)h(the)g Fv(M)1813 2125 y Fr(i)1882 2110 y Fx(and)g(the)h(homomorphisms)g Fv(\023)3023 2125 y Fr(j)3101 2110 y Fx(:)f Fv(M)3262 2125 y Fr(j)3339 2110 y Fq(\000)-59 b(!)3497 2035 y Fl(`)3607 2110 y Fv(M)3701 2125 y Fr(i)3770 2110 y Fx(are)0 2226 y(called)39 b Fs(inje)-5 b(ctions)p Fx(,)39 b(if)f(for)g(eac)m(h)h Fv(R)q Fx(-mo)s(dule)g Fv(N)49 b Fx(and)38 b(for)g(eac)m(h)h(family)g (of)f(homomorphisms)i(\()p Fv(f)3630 2241 y Fr(j)3704 2226 y Fx(:)e Fv(M)3863 2241 y Fr(j)0 2342 y Fq(\000)-60 b(!)28 b Fv(N)10 b Fq(j)p Fv(j)34 b Fq(2)28 b Fv(I)8 b Fx(\))32 b(there)h(is)g(a)g(unique)h(homomorphism)f Fv(f)39 b Fx(:)2126 2268 y Fl(`)2236 2342 y Fv(M)2330 2357 y Fr(j)2395 2342 y Fq(\000)-60 b(!)28 b Fv(N)43 b Fx(suc)m(h)34 b(that)1611 2556 y Fv(M)1705 2571 y Fr(j)2056 2481 y Fl(`)2166 2556 y Fv(M)2260 2571 y Fr(i)p 1771 2534 248 4 v 1936 2532 a Fj(-)1859 2485 y Fv(\023)1893 2500 y Fr(j)1775 2797 y Fv(f)1823 2812 y Fr(j)1745 2684 y Fj(@)1828 2767 y(@)1911 2850 y(@)1994 2933 y(@)2013 2952 y(@)-83 b(R)2120 3054 y Fv(N)p 2162 2952 4 351 v 2164 2952 a Fj(?)2203 2801 y Fv(f)0 3200 y Fx(comm)m(ute)34 b(for)e(all)h Fv(j)g Fq(2)28 b Fv(I)8 b Fx(.)0 3388 y Ft(Remark)51 b(3.2.)c Fx(An)d(analogous)g(de\014nition)h(can)f(b)s(e)g (giv)m(en)g(for)g(algebras,)j(coalgebras,)g(como)s(dules,)0 3504 y(groups,)33 b(Ab)s(elian)g(groups)g(etc.)0 3692 y Fs(Note:)67 b Fx(If)45 b(y)m(ou)g(w)m(an)m(t)g(to)f(de\014ne)i(a)e (homomorphism)h Fv(f)59 b Fx(:)48 b Fv(N)58 b Fq(\000)-59 b(!)2553 3617 y Fl(Q)2664 3692 y Fv(M)2758 3707 y Fr(i)2831 3692 y Fx(with)45 b(a)f(pro)s(duct)g(as)h(range)0 3808 y(\(co)s(domain\))33 b(y)m(ou)g(should)g(de\014ne)h(it)f(b)m(y)g (giving)g(homomorphisms)h Fv(f)2591 3823 y Fr(i)2647 3808 y Fx(:)28 b Fv(N)38 b Fq(\000)-59 b(!)27 b Fv(M)3057 3823 y Fr(i)3086 3808 y Fx(.)0 3925 y(If)41 b(y)m(ou)g(w)m(an)m(t)h(to) e(de\014ne)i(a)e(homomorphism)i Fv(f)52 b Fx(:)1917 3850 y Fl(`)2028 3925 y Fv(M)2122 3940 y Fr(i)2192 3925 y Fq(\000)-60 b(!)41 b Fv(N)52 b Fx(with)41 b(a)f(copro)s(duct)h(as)g (domain)g(y)m(ou)0 4041 y(should)33 b(de\014ne)h(it)f(b)m(y)g(giving)g (homomorphisms)i Fv(f)1890 4056 y Fr(i)1945 4041 y Fx(:)28 b Fv(M)2094 4056 y Fr(i)2150 4041 y Fq(\000)-59 b(!)27 b Fv(N)10 b Fx(.)0 4229 y Ft(Lemma)39 b(3.3.)j Fs(Pr)-5 b(o)g(ducts)35 b(and)f(c)-5 b(opr)g(o)g(ducts)35 b(ar)-5 b(e)34 b(unique)h(up)g(to)g(a)g(unique)g(isomorphism.)0 4416 y(Pr)-5 b(o)g(of.)41 b Fx(analogous)32 b(to)h(Prop)s(osition1.6.) 2311 b Fo(\003)0 4604 y Ft(Prop)s(osition)38 b(3.4.)k Fx(\(Rules)34 b(of)e(computation)h(in)h(a)e(pro)s(duct)h(of)g Fv(R)q Fx(-mo)s(dules\))i Fs(L)-5 b(et)36 b Fx(\()3219 4529 y Fl(Q)3330 4604 y Fv(M)3424 4619 y Fr(i)3452 4604 y Fv(;)17 b Fx(\()p Fv(p)3583 4619 y Fr(j)3619 4604 y Fx(\)\))35 b Fs(b)-5 b(e)35 b(a)0 4720 y(pr)-5 b(o)g(duct)35 b(of)g(the)f(family)h(of)f Fv(R)q Fs(-mo)-5 b(dules)34 b Fx(\()p Fv(M)1645 4735 y Fr(i)1674 4720 y Fx(\))1712 4735 y Fr(i)p Fp(2)p Fr(I)1823 4720 y Fs(.)148 4869 y Fx(\(1\))42 b Fs(Ther)-5 b(e)34 b(is)g(a)h(bije)-5 b(ction)34 b(of)h(sets)812 4976 y Fl(Y)956 5071 y Fv(M)1050 5086 y Fr(i)1106 5071 y Fq(3)28 b Fv(a)g Fq(7!)f Fx(\()p Fv(a)1495 5086 y Fr(i)1524 5071 y Fx(\))g(:=)h(\()p Fv(p)1807 5086 y Fr(i)1835 5071 y Fx(\()p Fv(a)p Fx(\)\))g Fq(2)2122 4990 y Fl(\010)2180 5071 y Fx(\()p Fv(a)2269 5086 y Fr(i)2297 5071 y Fx(\))p Fq(j8)p Fv(i)g Fq(2)g Fv(I)36 b Fx(:)27 b Fv(a)2757 5086 y Fr(i)2813 5071 y Fq(2)i Fv(M)3002 5086 y Fr(i)3030 4990 y Fl(\011)315 5273 y 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Fr(i)429 5622 y Fx(\()p Fv(n)p Fx(\)\))p Fs(.)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 1367 -170 a Fu(Pro)t(jectiv)n(e)27 b(Mo)r(dules)g(and)e(Generators)1266 b(31)0 29 y Fs(Pr)-5 b(o)g(of.)41 b Fx(Let)36 b(a)f(family)h(\()p Fv(a)950 44 y Fr(i)979 29 y Fq(j)p Fv(i)d Fq(2)g Fv(I)8 b Fx(\))35 b(b)s(e)h(giv)m(en.)54 b(F)-8 b(orm)35 b Fv(')2057 44 y Fr(i)2118 29 y Fx(:)e Fq(f)p Fx(1)p Fq(g)f(\000)-59 b(!)32 b Fv(M)2603 44 y Fr(i)2667 29 y Fx(with)37 b Fv(')2957 44 y Fr(i)2985 29 y Fx(\(1\))32 b(=)h Fv(a)3302 44 y Fr(i)3366 29 y Fx(for)i(all)h Fv(i)d Fq(2)g Fv(I)8 b Fx(.)0 146 y(Construct)34 b Fv(g)503 161 y Fr(i)558 146 y Fq(2)28 b Fx(Hom)856 161 y Fr(R)913 146 y Fx(\()p Fv(R)q(;)17 b(M)1164 161 y Fr(i)1193 146 y Fx(\))32 b(suc)m(h)i(that)e(the)h (diagrams)1613 296 y Fq(f)p Fx(1)p Fq(g)375 b Fv(R)p 1790 273 318 4 v 2025 271 a Fj(-)1778 526 y Fv(')1842 541 y Fr(i)1755 422 y Fj(@)1838 505 y(@)1921 588 y(@)2004 671 y(@)2023 690 y(@)-83 b(R)2113 785 y Fv(M)2207 800 y Fr(i)p 2173 690 4 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Fj(-)1956 2399 y Fl(Q)2066 2474 y Fv(M)2160 2489 y Fr(i)1653 2113 y Fj(@)1736 2196 y(@)1819 2279 y(@)1902 2362 y(@)1921 2381 y(@)-83 b(R)2007 2957 y Fv(M)2101 2972 y Fr(j)1669 2457 y Fv(')1733 2472 y Fr(j)1619 2113 y Fj(A)1661 2196 y(A)1702 2279 y(A)1744 2362 y(A)1785 2445 y(A)1827 2528 y(A)1868 2611 y(A)1910 2694 y(A)1951 2777 y(A)1993 2860 y(A)1997 2868 y(A)-42 b(U)p 2071 2381 4 351 v 2072 2381 a(?)1983 2217 y Fv(g)p 2071 2868 V 2072 2868 a Fj(?)1948 2700 y Fv(p)1997 2715 y Fr(j)2141 2113 y Fj(@)2175 2147 y(@)-83 b(R)p 2266 2741 4 585 v 2267 2741 a(?)2306 2457 y Fv(g)2353 2472 y Fr(j)2175 2844 y Fj(\000)2141 2878 y(\000)g(\011)0 3089 y Fx(where)35 b Fv(p)332 3104 y Fr(j)369 3089 y Fx(\()p Fv(a)p Fx(\))30 b(=)g Fv(')696 3104 y Fr(j)732 3089 y Fx(\(1\))g(=)g Fv(a)1044 3104 y Fr(j)1081 3089 y Fx(.)48 b(So)34 b(w)m(e)h(ha)m(v)m(e) g(found)g Fv(a)30 b Fq(2)2118 3015 y Fl(Q)2228 3089 y Fv(M)2322 3104 y Fr(i)2385 3089 y Fx(with)k(\()p Fv(p)2695 3104 y Fr(i)2724 3089 y Fx(\()p Fv(a)p 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3438 y Fx(are)k(equal)h(so)g(that)f Fv(g)f Fx(=)c Fv(h)33 b Fx(and)f(hence)i Fv(a)27 b Fx(=)h Fv(g)t Fx(\(1\))f(=)g Fv(h)p Fx(\(1\))h(=)f Fv(b)p Fx(.)44 b(Hence)34 b(the)0 3554 y(map)f(giv)m(en)g(in)g(the)g(prop)s(osition)g(is)g(bijectiv)m(e.) 0 3670 y(Since)f Fv(a)f Fx(is)h(uniquely)h(determined)g(b)m(y)e(the)h Fv(p)1683 3685 y Fr(j)1719 3670 y Fx(\()p Fv(a)p Fx(\))c(=)g Fv(a)2029 3685 y Fr(j)2096 3670 y Fx(w)m(e)k(ha)m(v)m(e)h Fv(p)2511 3685 y Fr(j)2547 3670 y Fx(\()p Fv(a)19 b Fx(+)g Fv(b)p Fx(\))28 b(=)f Fv(p)3009 3685 y Fr(j)3046 3670 y Fx(\()p Fv(a)p Fx(\))19 b(+)g Fv(p)3336 3685 y Fr(j)3372 3670 y Fx(\()p Fv(b)p Fx(\))28 b(=)g Fv(a)3672 3685 y Fr(j)3727 3670 y Fx(+)19 b Fv(b)3863 3685 y Fr(j)0 3787 y Fx(and)33 b Fv(p)239 3802 y Fr(j)275 3787 y Fx(\()p Fv(r)s(a)p Fx(\))28 b(=)f Fv(r)s(p)676 3802 y Fr(j)712 3787 y Fx(\()p Fv(a)p Fx(\))h(=)g Fv(r)s(a)1069 3802 y Fr(j)0 3903 y Fx(The)34 b(last)e(statemen)m(t)i(is)f Fv(p)985 3918 y Fr(i)1014 3903 y Fv(f)38 b Fx(=)28 b Fv(f)1252 3918 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b(3.8)e(w)m(e)h(ha)m(v)m(e)h(an)f(in)m(ternal)g(direct)g (sum.)1275 b Fo(\003)0 5517 y Ft(De\014nition)55 b(3.11.)50 b Fx(A)e(submo)s(dule)h Fv(M)64 b Fq(\022)53 b Fv(N)58 b Fx(is)48 b(called)h(a)e Fs(dir)-5 b(e)g(ct)49 b(summand)d Fx(of)h Fv(N)58 b Fx(if)48 b(there)g(is)g(a)0 5633 y(submo)s(dule)34 b Fv(M)595 5597 y Fp(0)647 5633 y Fq(\022)28 b Fv(N)43 b Fx(suc)m(h)34 b(that)e Fv(N)39 b Fx(=)27 b Fv(M)33 b Fq(\010)23 b Fv(M)1855 5597 y Fp(0)1911 5633 y Fx(is)33 b(an)g(in)m(ternal)g(direct)h(sum.)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 0 -170 a Fu(34)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Ft(Prop)s(osition)38 b(3.12.)j Fs(F)-7 b(or)34 b(a)h(submo)-5 b(dule)34 b Fv(M)39 b Fq(\022)28 b Fv(N)45 b Fs(the)35 b(fol)5 b(lowing)34 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)148 173 y Fx(\(1\))42 b Fv(M)j Fs(is)35 b(a)g(dir)-5 b(e)g(ct)34 b(summand)g(of)h Fv(N)10 b Fs(.)148 289 y Fx(\(2\))42 b Fs(Ther)-5 b(e)34 b(is)g Fv(p)28 b Fq(2)g Fx(Hom)1073 304 y Fr(R)1131 289 y Fx(\()p 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Fp(\003)2492 5109 y Fq(\012)2569 5124 y Fr(R)2634 5109 y Fv(P)41 b Fq(\000)-60 b(!)27 b Fv(R)q Fx(\).)41 b(Let)25 b(db)3338 5066 y Fp(0)3362 5109 y Fx(\(1\))i(=)3618 5034 y Fl(P)3739 5109 y Fv(p)3788 5124 y Fr(i)3823 5109 y Fq(\012)0 5225 y Fv(f)48 5240 y Fr(i)98 5225 y Fx(b)s(e)22 b(the)g(dual)g(basis)h(for) e Fv(P)14 b Fx(.)39 b(Then)23 b(w)m(e)g(ha)m(v)m(e)g(\(id)17 b Fq(\012)1897 5240 y Fr(R)1972 5225 y Fx(ev)r(\)\(db)2252 5182 y Fp(0)2292 5225 y Fq(\012)2369 5240 y Fr(R)2444 5225 y Fx(id\)\()p Fv(p)p Fx(\))28 b(=)f(\(id)17 b Fq(\012)3032 5240 y Fr(R)3107 5225 y Fx(ev)r(\)\()3279 5150 y Fl(P)3401 5225 y Fv(p)3450 5240 y Fr(i)3478 5225 y Fq(\012)p Fv(f)3603 5240 y Fr(i)3632 5225 y Fq(\012)p Fv(p)p Fx(\))28 b(=)0 5267 y Fl(P)122 5341 y Fv(p)171 5356 y Fr(i)199 5341 y Fv(f)247 5356 y Fr(i)275 5341 y Fx(\()p Fv(p)p Fx(\))g(=)f Fv(p)p Fx(.)41 b(F)-8 b(urthermore)24 b(w)m(e)i(ha)m(v)m(e)1556 5267 y Fl(P)1678 5341 y Fv(f)11 b Fx(\()p Fv(p)1824 5356 y Fr(i)1852 5341 y Fx(\))p Fv(f)1938 5356 y Fr(i)1966 5341 y Fx(\()p Fv(p)p Fx(\))27 b(=)h Fv(f)11 b Fx(\()2319 5267 y Fl(P)2440 5341 y Fv(p)2489 5356 y Fr(i)2517 5341 y Fv(f)2565 5356 y Fr(i)2594 5341 y Fx(\()p Fv(p)p Fx(\)\))27 b(=)h Fv(f)11 b Fx(\()p Fv(p)p Fx(\),)25 b(hence)3387 5267 y Fl(P)3508 5341 y Fv(f)11 b Fx(\()p Fv(p)3654 5356 y Fr(i)3682 5341 y Fx(\))p Fv(f)3768 5356 y Fr(i)3824 5341 y Fx(=)0 5458 y Fv(f)g Fx(.)43 b(This)34 b(implies)g(\(ev)19 b Fq(\012)911 5473 y Fr(R)986 5458 y Fx(id\)\(id)e Fq(\012)1318 5473 y Fr(R)1392 5458 y Fx(db)1501 5415 y Fp(0)1524 5458 y Fx(\)\()p Fv(f)11 b Fx(\))27 b(=)h(\(ev)19 b Fq(\012)2056 5473 y Fr(R)2130 5458 y Fx(id)q(\)\()2288 5383 y Fl(P)2409 5458 y Fv(f)33 b Fq(\012)23 b Fv(p)2639 5473 y Fr(i)2689 5458 y Fq(\012)g Fv(f)2837 5473 y Fr(i)2865 5458 y Fx(\))28 b(=)3034 5383 y Fl(P)3156 5458 y Fv(f)11 b Fx(\()p Fv(p)3302 5473 y Fr(i)3330 5458 y Fx(\))p Fv(f)3416 5473 y Fr(i)3472 5458 y Fx(=)27 b Fv(f)11 b Fx(.)162 b Fo(\003)0 5646 y Ft(Example)39 b(3.22.)i Fx(of)32 b(a)h(pro)5 b(jectiv)m(e)34 b(mo)s(dule,)g(that)e(is)h(not)f(free:)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 0 -170 a Fu(38)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fx(Let)j Fv(S)237 -7 y Fk(2)304 29 y Fx(=)g(2-sphere)g(=)f Fq(f)p Fx(\()p Fv(x)1037 44 y Fk(1)1077 29 y Fv(;)17 b(x)1176 44 y Fk(2)1215 29 y Fv(;)g(x)1314 44 y Fk(3)1354 29 y Fx(\))27 b Fq(2)i Fn(R)1586 -7 y Fk(3)1625 29 y Fq(j)p Fv(x)1708 -7 y Fk(2)1708 54 y(1)1761 29 y Fx(+)14 b Fv(x)1906 -7 y Fk(2)1906 54 y(2)1959 29 y Fx(+)g Fv(x)2104 -7 y Fk(2)2104 54 y(3)2172 29 y Fx(=)27 b(1)p Fq(g)p Fx(.)42 b(Let)28 b Fv(R)i Fx(b)s(e)f(the)f (ring)h(of)f(all)g(con)m(tin)m(uous)0 146 y(real-v)-5 b(alued)34 b(functions)g(on)g Fv(S)1121 110 y Fk(2)1160 146 y Fx(.)46 b(Let)34 b Fv(F)43 b Fx(=)29 b Fq(f)p Fv(f)40 b Fx(:)29 b Fv(S)1880 110 y Fk(2)1949 146 y Fq(\000)-60 b(!)29 b Fn(R)2167 110 y Fk(3)2206 146 y Fq(j)p Fv(f)43 b Fx(con)m(tin)m(uous)r Fq(g)29 b Fx(=)g Fq(f)p Fx(\()p Fv(f)3101 161 y Fk(1)3141 146 y Fv(;)17 b(f)3233 161 y Fk(2)3272 146 y Fv(;)g(f)3364 161 y Fk(3)3403 146 y Fx(\))p Fq(j)p Fv(f)3517 161 y Fr(i)3574 146 y Fq(2)30 b Fv(R)q Fq(g)f Fx(=)0 262 y Fv(R)75 226 y Fk(3)157 262 y Fx(b)s(e)43 b(the)g(free)g Fv(R)q Fx(-mo)s(dule)g(on)g(three)g (generators.)75 b Fv(F)56 b Fx(is)43 b(a)f(set)i(of)e(v)m(ector)i(v)-5 b(alued)43 b(functions,)k(the)0 378 y(v)m(ectors)35 b(starting)e(in)g (the)g(p)s(oin)m(t)g(of)g Fv(S)1411 342 y Fk(2)1483 378 y Fx(where)h(their)f(coun)m(terimage)i(is.)45 b(These)35 b(are)e(v)m(ector)h(\014elds)g(o)m(v)m(er)0 494 y Fv(S)66 458 y Fk(2)105 494 y Fx(.)43 b(Let)30 b Fv(P)41 b Fx(=)28 b Fq(f)p Fx(tangen)m(tial)k(v)m(ector)i(\014elds)q Fq(g)c Fx(and)g Fv(Q)e Fx(=)g Fq(f)p Fx(normal)k(v)m(ector)i(\014elds)q Fq(g)p Fx(.)43 b(Then)31 b Fv(F)41 b Fx(=)28 b Fv(P)i Fq(\010)18 b Fv(Q)30 b Fx(as)0 611 y Fv(R)q Fx(-mo)s(dules.)51 b(So)35 b Fv(P)48 b Fx(and)35 b Fv(Q)g Fx(are)g(pro)5 b(jectiv)m(e.)52 b(F)-8 b(urthermore)35 b Fv(Q)2426 583 y Fq(\030)2427 615 y Fx(=)2535 611 y Fv(R)q Fx(.)50 b(Hence)36 b Fv(F)3087 583 y Fq(\030)3088 615 y Fx(=)3196 611 y Fv(P)h Fq(\010)25 b Fv(R)q Fx(.)50 b(Supp)s(ose)0 727 y Fv(P)d Fx(w)m(ere)35 b(free.)47 b(Ev)-5 b(aluating)34 b(all)g(elemen)m(ts)i(of)d Fv(P)47 b Fx(in)34 b(a)g(giv)m(en)g(p)s(oin) m(t)g Fv(p)c Fq(2)g Fv(S)2766 691 y Fk(2)2839 727 y Fx(w)m(e)k(get)g (the)g(tangen)m(t)g(plane)0 843 y(at)e Fv(p)h Fx(whic)m(h)h(is)f Fn(R)650 807 y Fk(2)689 843 y Fx(.)44 b(If)32 b Fv(P)46 b Fx(is)33 b(free)g(then)g(it)g(has)g(a)f(basis)i Fv(e)2111 858 y Fk(1)2150 843 y Fv(;)17 b(e)2239 858 y Fk(2)2311 843 y Fx(\(see)34 b(later)e(remarks)i(on)f(the)g(rank)f(of)h(free)0 959 y(mo)s(dules)39 b(o)m(v)m(er)h(a)e(comm)m(utativ)m(e)j(ring\).)61 b(F)-8 b(or)37 b Fv(p)h Fq(2)g Fv(S)2010 923 y Fk(2)2088 959 y Fx(w)m(e)h(ha)m(v)m(e)h Fv(e)2513 974 y Fk(1)2553 959 y Fx(\()p Fv(p)p Fx(\))p Fv(;)17 b(e)2767 974 y Fk(2)2806 959 y Fx(\()p Fv(p)p Fx(\))38 b(generates)i(the)f(tangen)m(t)0 1076 y(plane,)45 b(hence)e(is)g(a)e(basis)i(for)f(the)g(tangen)m(t)g (plane.)73 b(So)42 b Fv(e)2243 1091 y Fk(1)2282 1076 y Fx(\()p Fv(p)p Fx(\))i Fq(6)p Fx(=)f(0)f(for)f(all)h Fv(p)i Fq(2)g Fv(S)3233 1039 y Fk(2)3272 1076 y Fx(.)71 b(By)43 b(the)f(\\egg)0 1192 y(theorem")33 b(this)g(is)g(imp)s (ossible.)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 1367 -170 a Fu(Pro)t(jectiv)n(e)27 b(Mo)r(dules)g(and)e(Generators)1266 b(39)0 29 y Fx(3.4.)49 b Ft(Generators.)0 204 y(De\014nition)i(3.23.)d Fx(A)c(righ)m(t)g Fv(R)q Fx(-mo)s(dule)g Fv(G)1718 219 y Fr(R)1820 204 y Fx(is)g(called)h(a)e Fs(gener)-5 b(ator)44 b Fx(if)f(for)h(eac)m(h)h (homomorphism)0 320 y Fv(f)38 b Fx(:)28 b Fv(M)39 b Fq(\000)-60 b(!)27 b Fv(N)43 b Fx(with)34 b Fv(f)k Fq(6)p Fx(=)28 b(0)k(there)h(exists)h(a)f(homomorphism)h Fv(g)d Fx(:)c Fv(G)h Fq(\000)-60 b(!)28 b Fv(M)43 b Fx(suc)m(h)34 b(that)e Fv(f)11 b(g)31 b Fq(6)p Fx(=)d(0.)0 494 y Ft(Prop)s(osition)38 b(3.24.)j Fs(L)-5 b(et)36 b Fv(G)1123 509 y Fr(R)1215 494 y Fs(b)-5 b(e)35 b(an)f Fv(R)q Fs(-mo)-5 b(dule.)44 b(The)35 b(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent)148 631 y Fx(\(1\))42 b Fv(G)34 b Fs(is)h(a)g(gener)-5 b(ator,)148 747 y Fx(\(2\))42 b Fs(for)34 b(e)-5 b(ach)34 b Fv(R)q Fs(-mo)-5 b(dule)35 b Fv(M)1225 762 y Fr(R)1318 747 y Fs(ther)-5 b(e)34 b(is)h(a)g(set)g Fv(I)42 b Fs(and)35 b(an)f(epimorphism)f Fv(h)28 b Fx(:)3034 672 y Fl(`)3128 776 y Fr(I)3184 747 y Fv(G)g Fq(\000)-57 b(!)27 b Fv(M)10 b Fs(,)148 863 y Fx(\(3\))42 b Fv(R)35 b Fs(is)g(isomorphic)e(to)j(a)e (dir)-5 b(e)g(ct)35 b(summand)f(of)2053 789 y Fl(`)2147 892 y Fr(I)2204 863 y Fv(G)h Fs(\(for)f(an)h(appr)-5 b(opriate)34 b(set)h Fv(I)8 b Fs(\),)148 980 y Fx(\(4\))42 b Fs(ther)-5 b(e)34 b(ar)-5 b(e)35 b Fv(f)771 995 y Fk(1)811 980 y Fv(;)17 b(:)g(:)g(:)e(;)i(f)1077 995 y Fr(n)1152 980 y Fq(2)28 b Fv(G)1323 943 y Fp(\003)1390 980 y Fx(=)f(Hom)1697 995 y Fr(R)1754 980 y Fx(\()p Fv(G:;)17 b(R)q(:)p Fx(\))35 b Fs(and)f Fv(q)2347 995 y Fk(1)2387 980 y Fv(;)17 b(:)g(:)g(:)f(;)h(q) 2649 995 y Fr(n)2723 980 y Fq(2)29 b Fv(G)34 b Fs(with)3141 905 y Fl(P)3263 980 y Fv(f)3311 995 y Fr(i)3339 980 y Fx(\()p Fv(q)3420 995 y Fr(i)3448 980 y Fx(\))28 b(=)g(1)p Fs(.)0 1154 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))32 b(=)-17 b Fq(\))33 b Fx(\(2\):)43 b(De\014ne)33 b Fv(I)i Fx(:=)28 b(Hom)1556 1169 y Fr(R)1614 1154 y Fx(\()p Fv(G:;)17 b(M)5 b(:)p Fx(\).)44 b(Then)33 b(the)g(diagram)1551 1340 y Fv(G)28 b Fx(=)f Fv(G)1836 1355 y Fr(f)2076 1266 y Fl(`)2170 1369 y Fr(I)2226 1340 y Fv(G)2303 1355 y Fr(f)p 1911 1319 129 4 v 1956 1317 a Fj(-)1935 1270 y Fv(\023)1969 1285 y Fr(f)1841 1586 y Fv(f)1785 1469 y Fj(@)1868 1552 y(@)1951 1635 y(@)2034 1718 y(@)2053 1737 y(@)-83 b(R)2152 1839 y Fv(M)p 2202 1737 4 351 v 2204 1737 a Fj(?)2243 1596 y Fv(h)0 1962 y Fx(de\014nes)40 b(a)e(unique)i(homomorphism)f Fv(h)g Fx(with)f Fv(h\023)1842 1977 y Fr(f)1926 1962 y Fx(=)f Fv(f)49 b Fx(for)38 b(all)g Fv(f)48 b Fq(2)38 b Fv(I)8 b Fx(.)60 b(Let)38 b Fv(N)48 b Fx(=)38 b(Im\()p Fv(h)p Fx(\).)61 b(Consider)0 2078 y Fv(\027)38 b Fx(:)32 b Fv(M)43 b Fq(\000)-59 b(!)31 b Fv(M)5 b(=)-5 b(N)10 b Fx(.)51 b(If)35 b Fv(N)43 b Fq(6)p Fx(=)31 b Fv(M)46 b Fx(then)36 b Fv(\027)i Fq(6)p Fx(=)32 b(0.)51 b(Since)36 b Fv(G)f Fx(is)h(a)e(generator)h(there)h (exists)h(an)e Fv(f)46 b Fx(suc)m(h)36 b(that)0 2194 y Fv(\027)6 b(f)42 b Fq(6)p Fx(=)31 b(0.)49 b(This)35 b(implies)h Fv(\027)6 b(h)32 b Fq(6)p Fx(=)e(0)35 b(a)f(con)m (tradiction)h(to)f Fv(N)41 b Fx(=)31 b(Im)q(\()p Fv(h)p Fx(\).)49 b(Hence)35 b Fv(N)42 b Fx(=)31 b Fv(M)45 b Fx(so)34 b(that)h Fv(h)f Fx(is)h(an)0 2311 y(epimorphism.)0 2427 y(\(2\))f(=)-17 b Fq(\))34 b Fx(\(3\):)46 b(Let)727 2352 y Fl(`)838 2427 y Fv(G)30 b Fq(\000)-59 b(!)30 b Fv(R)35 b Fx(b)s(e)g(an)f(epimorphism.)50 b(Since)36 b Fv(R)f Fx(is)g(a)f(free)g(mo)s(dule)h(hence)h(pro)5 b(jectiv)m(e,)0 2543 y(3.16)32 b(implies)i(that)e Fv(R)i Fx(is)f(a)f(direct)i(summand)g(of)1868 2468 y Fl(`)1979 2543 y Fv(G)e Fx(up)h(to)f(isomorphism.)0 2659 y(\(3\))k(=)-17 b Fq(\))36 b Fx(\(4\):)50 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Fx(\)\()p Fv(g)t Fx(\))31 b(=)g Fv(f)11 b(g)40 b Fs(is)d(a)315 927 y(c)-5 b(ovariant)34 b(functor)h Fx(Mor)1258 942 y Fp(C)1303 927 y Fx(\()p Fv(X)r(;)17 b Fs(-)p Fx(\))p Fs(.)148 1043 y Fx(\(2\))42 b Fs(L)-5 b(et)35 b Fv(X)g Fq(2)28 b(C)6 b Fs(.)45 b(Then)1219 1209 y Fx(Ob)17 b Fq(C)34 b(3)28 b Fv(A)g Fq(7!)f Fx(Mor)1950 1224 y Fp(C)1995 1209 y Fx(\()p Fv(A;)17 b(X)8 b Fx(\))27 b Fq(2)i Fx(Ob)16 b(Set)529 1383 y(Mor)705 1398 y Fp(C)750 1383 y Fx(\()p Fv(A;)h(B)5 b Fx(\))28 b Fq(3)g Fv(f)38 b Fq(7!)27 b Fx(Mor)1533 1398 y Fp(C)1579 1383 y Fx(\()p Fv(f)5 b(;)17 b(X)8 b Fx(\))27 b Fq(2)h Fx(Mor)2138 1398 y Fk(Set)2240 1383 y Fx(\(Mor)2454 1398 y Fp(C)2499 1383 y Fx(\()p Fv(B)5 b(;)17 b(X)8 b Fx(\))p Fv(;)17 b Fx(Mor)3007 1398 y Fp(C)3052 1383 y Fx(\()p Fv(A;)g(X)8 b Fx(\)\))315 1528 y Fs(with)36 b Fx(Mor)704 1543 y Fp(C)749 1528 y Fx(\()p Fv(f)5 b(;)17 b(X)8 b Fx(\))31 b(:)g(Mor)1276 1543 y Fp(C)1321 1528 y Fx(\()p Fv(B)5 b(;)17 b(X)8 b Fx(\))30 b Fq(3)h Fv(g)j Fq(7!)d Fv(g)t(f)41 b Fq(2)31 b Fx(Mor)2361 1543 y Fp(C)2406 1528 y Fx(\()p Fv(A;)17 b(X)8 b Fx(\))36 b Fs(or)h Fx(Mor)3028 1543 y Fp(C)3073 1528 y Fx(\()p Fv(f)5 b(;)17 b(X)8 b Fx(\)\()p Fv(g)t Fx(\))30 b(=)g Fv(g)t(f)47 b Fs(is)36 b(a)315 1645 y(c)-5 b(ontr)g(avariant)34 b(functor)h Fx(Mor)1431 1660 y Fp(C)1476 1645 y Fx(\()p Fs(-)p Fv(;)17 b(X)8 b Fx(\))p Fs(.)0 1824 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))59 b(Mor)658 1839 y Fp(C)703 1824 y Fx(\()p Fv(X)r(;)17 b Fx(1)917 1839 y Fr(A)974 1824 y Fx(\)\()p Fv(g)t Fx(\))72 b(=)g(1)1408 1839 y Fr(A)1465 1824 y Fv(g)k Fx(=)c Fv(g)k Fx(=)d(id\()p Fv(g)t Fx(\))p Fv(;)17 b Fx(Mor)2435 1839 y Fp(C)2480 1824 y Fx(\()p Fv(X)r(;)g(f)11 b Fx(\))17 b(Mor)2935 1839 y Fp(C)2980 1824 y Fx(\()p Fv(X)r(;)g(g)t Fx(\)\()p Fv(h)p Fx(\))72 b(=)g Fv(f)11 b(g)t(h)72 b Fx(=)0 1940 y(Mor)176 1955 y Fp(C)221 1940 y Fx(\()p Fv(X)r(;)17 b(f)11 b(g)t Fx(\)\()p Fv(h)p Fx(\).)0 2057 y(\(2\))32 b(analogously)-8 b(.)3151 b Fo(\003)0 2236 y Ft(Remark)37 b(4.16.)k Fx(The)33 b(preceding)g(lemma)f(sho)m(ws)h(that)f(Mor)2312 2251 y Fp(C)2357 2236 y Fx(\(-)o Fv(;)17 b Fx(-\))31 b(is)h(a)g(functor)f(in)h(b)s(oth)g(argumen)m(ts.)0 2352 y(A)h(functor)h(in)f(t)m(w)m(o)h(argumen)m(ts)h(is)f(called)g(a)f Fs(bifunctor)p Fx(.)46 b(W)-8 b(e)33 b(can)h(regard)f(the)h(bifunctor)g (Mor)3550 2367 y Fp(C)3595 2352 y Fx(\(-)o Fv(;)17 b Fx(-\))33 b(as)0 2468 y(a)f(co)m(v)-5 b(arian)m(t)33 b(functor)1374 2593 y(Mor)1550 2608 y Fp(C)1595 2593 y Fx(\(-)p Fv(;)17 b Fx(-)o(\))27 b(:)h Fq(C)1920 2552 y Fr(op)2016 2593 y Fq(\002)23 b(C)34 b(\000)-60 b(!)28 b Fx(Set)17 b Fv(:)0 2739 y Fx(The)34 b(use)h(of)d(the)i(dual)g (category)f(remo)m(v)m(es)j(the)e(fact)f(that)g(the)h(bifunctor)f(Mor) 2975 2754 y Fp(C)3020 2739 y Fx(\(-)o Fv(;)17 b Fx(-)o(\))34 b(is)f(con)m(tra)m(v)-5 b(arian)m(t)0 2855 y(in)33 b(the)g(\014rst)g(v) -5 b(ariable.)0 2971 y(Ob)m(viously)38 b(the)e(comp)s(osition)h(of)e(t) m(w)m(o)i(functors)g(is)f(again)g(a)f(functor)h(and)g(this)h(comp)s (osition)g(is)f(asso-)0 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b Fx(\()p Fv(f)h Fx(\).)0 4964 y Ft(Lemma)39 b(4.18.)j Fs(Given)35 b(c)-5 b(ovariant)34 b(functors)h Fq(F)i Fx(=)27 b(Id)2063 4979 y Fk(Set)2193 4964 y Fx(:)g(Set)h Fq(\000)-57 b(!)28 b Fx(Set)35 b Fs(and)1078 5130 y Fq(G)f Fx(=)27 b(Mor)1450 5145 y Fk(Set)1553 5130 y Fx(\(Mor)1767 5145 y Fk(Set)1869 5130 y Fx(\()p Fq(\000)p Fv(;)17 b(A)p Fx(\))p Fv(;)g(A)p Fx(\))27 b(:)h(Set)g Fq(\000)-57 b(!)28 b Fx(Set)0 5296 y Fs(for)35 b(a)f(set)h Fv(A)p Fs(.)45 b(Then)34 b Fv(')28 b Fx(:)g Fq(F)37 b(\000)-57 b(!)27 b(G)41 b Fs(with)763 5461 y Fv(')p Fx(\()p Fv(B)5 b Fx(\))28 b(:)g Fv(B)33 b Fq(3)28 b Fv(b)g Fq(7!)f Fx(\(Mor)1676 5476 y Fk(Set)1778 5461 y Fx(\()p Fv(B)5 b(;)17 b(A)p Fx(\))28 b Fq(3)g Fv(f)38 b Fq(7!)28 b Fv(f)11 b Fx(\()p Fv(b)p Fx(\))27 b Fq(2)i Fv(A)p Fx(\))e Fq(2)h(G)6 b Fx(\()p Fv(B)f Fx(\))0 5627 y Fs(is)35 b(a)f(natur)-5 b(al)35 b(tr)-5 b(ansformation.)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 0 -170 a Fu(44)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fs(Pr)-5 b(o)g(of.)41 b Fx(Giv)m(en)34 b Fv(g)d Fx(:)c Fv(B)33 b Fq(\000)-59 b(!)27 b Fv(C)7 b Fx(.)43 b(Then)34 b(the)f(follo)m(wing)g(diagram)f (comm)m(utes)952 249 y Fv(B)438 b Fx(Mor)1640 264 y Fk(Set)1743 249 y Fx(\(Mor)1957 264 y Fk(Set)2059 249 y Fx(\()p Fv(B)5 b(;)17 b(A)p Fx(\))p Fv(;)g(A)p Fx(\))p 1060 225 368 4 v 1345 223 a Fj(-)1134 179 y Fv(')p Fx(\()p Fv(B)5 b Fx(\))953 736 y Fv(C)445 b Fx(Mor)1644 751 y Fk(Set)1746 736 y Fx(\(Mor)1960 751 y Fk(Set)2063 736 y Fx(\()p Fv(C)r(;)17 b(A)p Fx(\))p Fv(;)g(A)p Fx(\))p 1059 713 373 4 v 1349 711 a Fj(-)1137 667 y Fv(')p Fx(\()p Fv(C)7 b Fx(\))p 990 643 4 351 v 992 643 a Fj(?)902 479 y Fv(g)p 1965 643 V 1967 643 a Fj(?)2006 492 y Fx(Mor)2182 507 y Fk(Set)2284 492 y Fx(\(Mor)2498 507 y Fk(Set)2600 492 y Fx(\()p Fv(g)t(;)17 b(A)p Fx(\))p Fv(;)g(A)p Fx(\))0 883 y(since)782 980 y Fv(')p Fx(\()p Fv(C)7 b Fx(\))p Fq(F)j Fx(\()p Fv(g)t Fx(\)\()p Fv(b)p Fx(\)\()p Fv(f)h Fx(\))26 b(=)h Fv(')p Fx(\()p Fv(C)7 b Fx(\))p Fv(g)t Fx(\()p Fv(b)p 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b(that)e Fq(G)34 b Fx(:)28 b Fq(D)i(\000)-59 b(!)27 b(C)36 b Fx(is)30 b(an)f(equiv)-5 b(alence.)45 b(Sho)m(w)30 b(that)g Fq(G)35 b Fx(is)30 b(uniquely)i(determined)0 5310 y(b)m(y)h Fq(F)43 b Fx(up)32 b(to)h(a)f(natural)g(isomorphism.)p eop end %%Page: 46 46 TeXDict begin 46 45 bop 0 -170 a Fu(46)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)408 29 y Fx(5.)49 b Fw(Represent)-7 b(able)35 b(and)j(Adjoint)h(Functors,)e(the)g(Yoned)n(a)h(Lemma)0 204 y Fx(5.1.)49 b Ft(Represen)m(table)38 b(functors.)0 393 y(De\014nition)28 b(5.1.)35 b Fx(Let)24 b Fq(F)37 b Fx(:)28 b Fq(C)34 b(\000)-60 b(!)27 b Fx(Set)d(b)s(e)g(a)f(co)m(v)-5 b(arian)m(t)25 b(functor.)40 b(A)24 b(pair)f(\()p Fv(A;)17 b(x)p Fx(\))24 b(with)g Fv(A)k Fq(2)g(C)6 b Fv(;)17 b(x)28 b Fq(2)h(F)10 b Fx(\()p Fv(A)p Fx(\))0 509 y(is)46 b(called)h(a)e Fs(r)-5 b(epr)g(esenting)47 b(\(generic,)h(universal\))f(obje)-5 b(ct)45 b Fx(for)g Fq(F)55 b Fx(and)46 b Fq(F)56 b Fx(is)46 b(called)h(a)e Fs(r)-5 b(epr)g(esentable)0 626 y(functor)p 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b(represen)m(ting)j(ob)5 b(ject)27 b(for)f Fq(F)36 b Fx(is)28 b(giv)m(en)f(b)m(y)h(the)f(free)g Fv(R)q Fx(-mo)s(dule)0 3711 y(\()p Fv(R)q(X)r(;)17 b(x)34 b Fx(:)g Fv(X)41 b Fq(\000)-59 b(!)33 b Fv(R)q(X)8 b Fx(\))36 b(with)h(the)f(prop)s(ert)m(y)-8 b(,)38 b(that)e(for)f(all)h (\()p Fv(M)5 b(;)17 b(y)37 b Fx(:)d Fv(X)41 b Fq(\000)-59 b(!)33 b Fv(M)10 b Fx(\))37 b(there)g(exists)h(a)d(unique)0 3827 y Fv(f)j Fq(2)28 b Fx(Hom)384 3842 y Fr(R)441 3827 y Fx(\()p Fv(R)q(X)r(;)17 b(M)10 b Fx(\))34 b(suc)m(h)g(that)e Fq(F)10 b Fx(\()p Fv(f)h Fx(\)\()p Fv(x)p Fx(\))27 b(=)h(Map\()p Fv(X)r(;)17 b(f)11 b Fx(\)\()p Fv(x)p Fx(\))28 b(=)f Fv(f)11 b(x)28 b Fx(=)g Fv(y)1643 4008 y(X)369 b(R)q(X)p 1760 3975 304 4 v 1981 3973 a Fj(-)1892 3955 y Fr(x)1833 4225 y(y)1755 4125 y Fj(@)1839 4208 y(@)1922 4291 y(@)2005 4374 y(@)2023 4393 y(@)-83 b(R)2111 4495 y Fv(M)5 b(:)p 2173 4393 4 351 v 2175 4393 a Fj(?)2214 4234 y Fr(f)0 4644 y Fx(\(2\))25 b(Giv)m(en)g(mo)s(dules)h Fv(M)893 4659 y Fr(R)976 4644 y Fx(and)1158 4659 y Fr(R)1216 4644 y Fv(N)10 b Fx(.)42 b(De\014ne)25 b Fq(F)37 b Fx(:)28 b(Ab)g Fq(\000)-60 b(!)28 b Fx(Set)d(b)m(y)h Fq(F)10 b Fx(\()p Fv(A)p Fx(\))27 b(:=)h(Bil)2931 4659 y Fr(R)2989 4644 y Fx(\()p Fv(M)5 b(;)17 b(N)10 b Fx(;)17 b Fv(A)p Fx(\).)41 b(Then)26 b Fq(F)34 b Fx(is)0 4760 y(a)27 b(co)m(v)-5 b(arian)m(t)28 b(functor.)42 b(A)28 b(represen)m(ting)h(ob)5 b(ject)28 b(for)f Fq(F)37 b Fx(is)28 b(giv)m(en)h(b)m(y)f(the)g(tensor) g(pro)s(duct)g(\()p Fv(M)22 b Fq(\012)3571 4775 y Fr(R)3641 4760 y Fv(N)5 b(;)17 b Fq(\012)28 b Fx(:)0 4876 y Fv(M)38 b Fq(\002)28 b Fv(N)52 b Fq(\000)-60 b(!)41 b Fv(M)d Fq(\012)734 4891 y Fr(R)820 4876 y Fv(N)10 b Fx(\))41 b(with)g(the)f(prop)s(ert)m(y)i(that)e(for)g(all)g(\()p Fv(A;)17 b(f)52 b Fx(:)41 b Fv(M)d Fq(\002)28 b Fv(N)51 b Fq(\000)-59 b(!)40 b Fv(A)p Fx(\))h(there)g(exists)h(a)0 4993 y(unique)34 b Fv(g)d Fq(2)d Fx(Hom\()p Fv(M)33 b Fq(\012)934 5008 y Fr(R)1014 4993 y Fv(N)5 b(;)17 b(A)p Fx(\))33 b(suc)m(h)h(that)e Fq(F)10 b Fx(\()p Fv(g)t Fx(\)\()p Fq(\012)p Fx(\))27 b(=)g(Bil)2332 5008 y Fr(R)2389 4993 y Fx(\()p Fv(M)5 b(;)17 b(N)10 b Fx(;)17 b Fv(g)t Fx(\)\()p Fq(\012)p Fx(\))28 b(=)f Fv(g)t Fq(\012)h Fx(=)f Fv(f)1534 5186 y(M)33 b Fq(\002)23 b Fv(N)154 b(M)33 b Fq(\012)2197 5201 y Fr(R)2277 5186 y Fv(N)p 1878 5161 86 4 v 1881 5159 a Fj(-)1893 5133 y Fp(\012)1834 5419 y Fr(f)1760 5310 y Fj(@)1843 5393 y(@)1926 5476 y(@)2009 5559 y(@)2028 5578 y(@)-83 b(R)2129 5680 y Fv(A:)p 2177 5578 4 351 v 2179 5578 a Fj(?)2218 5411 y Fr(g)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 992 -170 a Fu(Represen)n(table)26 b(and)f(Adjoin)n(t)g(F)-6 b(unctors,)26 b(the)f(Y)-6 b(oneda)25 b(Lemma)916 b(47)0 29 y Fx(\(3\))41 b(Giv)m(en)h(a)f Fn(K)p Fx(-mo)s(dule)h Fv(V)22 b Fx(.)70 b(De\014ne)42 b Fq(F)52 b Fx(:)43 b Fn(K)p Fx(-Alg)g Fq(\000)-60 b(!)43 b Fx(Set)e(b)m(y)i Fq(F)10 b Fx(\()p Fv(A)p Fx(\))42 b(:=)h(Hom\()p Fv(V)5 b(;)17 b(A)p Fx(\).)70 b(Then)43 b Fq(F)50 b Fx(is)0 146 y(a)39 b(co)m(v)-5 b(arian)m(t)40 b(functor.)64 b(A)39 b(represen)m(ting)j(ob)5 b(ject)40 b(for)f Fq(F)49 b Fx(is)40 b(giv)m(en)g(b)m(y)g(the)g(tensor)g(algebra) f(\()p Fv(T)14 b Fx(\()p Fv(V)22 b Fx(\))p Fv(;)17 b(\023)39 b Fx(:)0 262 y Fv(V)72 b Fq(\000)-60 b(!)50 b Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\)\))46 b(with)g(the)h(prop)s(ert)m(y)f(that)g (for)f(all)h(\()p Fv(A;)17 b(f)61 b Fx(:)50 b Fv(V)72 b Fq(\000)-60 b(!)50 b Fv(A)p Fx(\))c(there)h(exists)g(a)f(unique)h Fv(g)54 b Fq(2)0 378 y Fx(Mor)176 393 y Fk(Alg)289 378 y Fx(\()p Fv(T)14 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(A)p Fx(\))32 b(suc)m(h)i(that)f Fq(F)10 b Fx(\()p Fv(g)t Fx(\)\()p Fv(\023)p Fx(\))26 b(=)i(Hom\()p Fv(V)5 b(;)17 b(g)t Fx(\)\()p Fv(\023)p Fx(\))27 b(=)h Fv(g)t(\023)f Fx(=)h Fv(f)1630 564 y(V)358 b(T)14 b Fx(\()p Fv(V)21 b Fx(\))p 1738 541 278 4 v 1933 539 a Fj(-)1864 520 y Fr(\023)1812 799 y(f)1738 690 y Fj(@)1821 773 y(@)1904 856 y(@)1987 939 y(@)2006 958 y(@)-83 b(R)2107 1060 y Fv(A:)p 2156 958 4 351 v 2157 958 a Fj(?)2196 791 y Fr(g)0 1218 y Fx(\(4\))36 b(Giv)m(en)h(a)f Fn(K)p Fx(-mo)s(dule)h Fv(V)22 b Fx(.)54 b(De\014ne)37 b Fq(F)44 b Fx(:)34 b Fn(K)p Fx(-cAlg)g Fq(\000)-59 b(!)34 b Fx(Set)i(b)m(y)i Fq(F)10 b Fx(\()p Fv(A)p Fx(\))33 b(:=)h(Hom\()p Fv(V)5 b(;)17 b(A)p Fx(\).)55 b(Then)38 b Fq(F)45 b Fx(is)37 b(a)0 1334 y(co)m(v)-5 b(arian)m(t)35 b(functor.)48 b(A)34 b(represen)m(ting)i(ob)5 b(ject)35 b(for)e Fq(F)44 b Fx(is)34 b(giv)m(en)i(b)m(y)f(the)f(symmetric)i(algebra)e(\()p Fv(S)6 b Fx(\()p Fv(V)22 b Fx(\))p Fv(;)17 b(\023)30 b Fx(:)0 1450 y Fv(V)72 b Fq(\000)-59 b(!)50 b Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\)\))46 b(with)h(the)f(prop)s(ert)m(y)h(that)f (for)f(all)h(\()p Fv(A;)17 b(f)61 b Fx(:)51 b Fv(V)72 b Fq(\000)-59 b(!)50 b Fv(A)p Fx(\))c(there)h(exists)g(a)f(unique)i Fv(g)54 b Fq(2)0 1566 y Fx(Mor)176 1581 y Fk(cAlg)320 1566 y Fx(\()p Fv(S)6 b Fx(\()p Fv(V)22 b Fx(\))p Fv(;)17 b(A)p Fx(\))32 b(suc)m(h)i(that)e Fq(F)10 b Fx(\()p Fv(g)t Fx(\)\()p Fv(\023)p Fx(\))27 b(=)g(Hom)q(\()p Fv(V)5 b(;)17 b(g)t Fx(\)\()p Fv(\023)p Fx(\))27 b(=)g Fv(g)t(\023)h Fx(=)f Fv(f)1632 1752 y(V)360 b(S)6 b Fx(\()p Fv(V)21 b Fx(\))p 1739 1729 281 4 v 1937 1727 a Fj(-)1867 1708 y Fr(\023)1813 1987 y(f)1739 1878 y Fj(@)1822 1961 y(@)1905 2044 y(@)1988 2127 y(@)2007 2146 y(@)-83 b(R)2109 2249 y Fv(A:)p 2157 2146 4 351 v 2159 2146 a Fj(?)2198 1979 y Fr(g)0 2406 y Fx(\(5\))32 b(Giv)m(en)i(a)e Fn(K)p Fx(-mo)s(dule)h Fv(V)22 b Fx(.)43 b(De\014ne)33 b Fq(F)k Fx(:)28 b Fn(K)p Fx(-Alg)g Fq(\000)-60 b(!)28 b Fx(Set)33 b(b)m(y)646 2597 y Fq(F)10 b Fx(\()p Fv(A)p Fx(\))27 b(:=)h Fq(f)p Fv(f)38 b Fq(2)28 b Fx(Hom)q(\()p Fv(V)5 b(;)17 b(A)p Fx(\))p Fq(j8)p Fv(v)t(;)g(v)1953 2556 y Fp(0)2003 2597 y Fq(2)28 b Fv(V)49 b Fx(:)28 b Fv(f)11 b Fx(\()p Fv(v)t Fx(\))p Fv(f)g Fx(\()p Fv(v)2592 2556 y Fp(0)2614 2597 y Fx(\))27 b(=)h Fv(f)11 b Fx(\()p Fv(v)2931 2556 y Fp(0)2954 2597 y Fx(\))p Fv(f)g Fx(\()p Fv(v)t Fx(\))p Fq(g)p Fv(:)0 2788 y Fx(Then)48 b Fq(F)56 b Fx(is)48 b(a)e(co)m(v)-5 b(arian)m(t)47 b(functor.)87 b(A)47 b(represen)m(ting)h(ob)5 b(ject)48 b(for)e Fq(F)57 b Fx(is)47 b(giv)m(en)h(b)m(y)g(the)f (symmetric)0 2904 y(algebra)42 b(\()p Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(\023)44 b Fx(:)g Fv(V)65 b Fq(\000)-59 b(!)43 b Fv(S)6 b Fx(\()p Fv(V)22 b Fx(\)\))42 b(with)g(the)h(prop)s (ert)m(y)g(that)e(for)h(all)g(\()p Fv(A;)17 b(f)54 b Fx(:)44 b Fv(V)66 b Fq(\000)-60 b(!)44 b Fv(A)p Fx(\))e(suc)m(h)h(that) 0 3020 y Fv(f)11 b Fx(\()p Fv(v)t Fx(\))p Fv(f)g Fx(\()p Fv(v)334 2984 y Fp(0)356 3020 y Fx(\))31 b(=)g Fv(f)11 b Fx(\()p Fv(v)680 2984 y Fp(0)703 3020 y Fx(\))p Fv(f)g Fx(\()p Fv(v)t Fx(\))33 b(for)h(all)h Fv(v)t(;)17 b(v)1395 2984 y Fp(0)1448 3020 y Fq(2)32 b Fv(V)56 b Fx(there)35 b(exists)i(a)d(unique)i Fv(g)e Fq(2)e Fx(Mor)2939 3035 y Fk(Alg)3052 3020 y Fx(\()p Fv(S)6 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(A)p Fx(\))34 b(suc)m(h)i(that)0 3137 y Fq(F)10 b Fx(\()p Fv(g)t Fx(\)\()p Fv(\023)p Fx(\))27 b(=)g(Hom\()p Fv(V)5 b(;)17 b(g)t Fx(\)\()p Fv(\023)p Fx(\))28 b(=)f Fv(g)t(\023)h Fx(=)f Fv(f)1632 3279 y(V)360 b(S)6 b Fx(\()p Fv(V)21 b Fx(\))p 1739 3256 281 4 v 1937 3254 a Fj(-)1867 3235 y Fr(\023)1813 3514 y(f)1739 3405 y Fj(@)1822 3488 y(@)1905 3571 y(@)1988 3654 y(@)2007 3673 y(@)-83 b(R)2109 3775 y Fv(A:)p 2157 3673 4 351 v 2159 3673 a Fj(?)2198 3505 y Fr(g)0 3912 y Fx(\(6\))32 b(Giv)m(en)i(a)e Fn(K)p Fx(-mo)s(dule)h Fv(V)22 b Fx(.)43 b(De\014ne)33 b Fq(F)k Fx(:)28 b Fn(K)p Fx(-Alg)g Fq(\000)-60 b(!)28 b Fx(Set)33 b(b)m(y)962 4104 y Fq(F)10 b Fx(\()p Fv(A)p Fx(\))27 b(:=)h Fq(f)p Fv(f)38 b Fq(2)28 b Fx(Hom)q(\()p Fv(V)5 b(;)17 b(A)p Fx(\))p Fq(j8)p Fv(v)31 b Fq(2)d Fv(V)49 b Fx(:)28 b Fv(f)11 b Fx(\()p Fv(v)t Fx(\))2642 4063 y Fk(2)2709 4104 y Fx(=)27 b(0)p Fq(g)p Fv(:)0 4295 y Fx(Then)35 b Fq(F)42 b Fx(is)34 b(a)f(co)m(v)-5 b(arian)m(t)34 b(functor.)46 b(A)34 b(represen)m(ting)h(ob)5 b(ject)35 b(for)d Fq(F)43 b Fx(is)34 b(giv)m(en)h(b)m(y)f(the)g(exterior)g (algebra)0 4412 y(\()p Fv(E)6 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(\023)35 b Fx(:)f Fv(V)56 b Fq(\000)-60 b(!)34 b Fv(E)6 b Fx(\()p Fv(V)22 b Fx(\)\))36 b(with)h(the)g(prop)s(ert)m(y)g (that)f(for)g(all)g(\()p Fv(A;)17 b(f)45 b Fx(:)35 b Fv(V)55 b Fq(\000)-59 b(!)34 b Fv(A)p Fx(\))i(suc)m(h)i(that)e Fv(f)11 b Fx(\()p Fv(v)t Fx(\))3668 4375 y Fk(2)3741 4412 y Fx(=)34 b(0)0 4528 y(for)29 b(all)g Fv(v)i Fq(2)d Fv(V)51 b Fx(there)30 b(exists)h(a)e(unique)i Fv(g)f Fq(2)f Fx(Mor)1810 4543 y Fk(Alg)1923 4528 y Fx(\()p Fv(E)6 b Fx(\()p Fv(V)21 b Fx(\))p Fv(;)c(A)p Fx(\))29 b(suc)m(h)i(that)e Fq(F)10 b Fx(\()p Fv(g)t Fx(\)\()p Fv(\023)p Fx(\))26 b(=)i(Hom\()p Fv(V)5 b(;)17 b(g)t Fx(\)\()p Fv(\023)p Fx(\))27 b(=)0 4644 y Fv(g)t(\023)g Fx(=)h Fv(f)1628 4781 y(V)354 b(E)6 b Fx(\()p Fv(V)22 b Fx(\))p 1736 4757 274 4 v 1927 4755 a Fj(-)1860 4737 y Fr(\023)1810 5016 y(f)1736 4907 y Fj(@)1819 4990 y(@)1902 5073 y(@)1985 5156 y(@)2004 5175 y(@)-83 b(R)2105 5277 y Fv(A:)p 2154 5175 4 351 v 2155 5175 a Fj(?)2194 5007 y Fr(g)0 5414 y Fx(\(7\))41 b(Let)h Fn(K)h Fx(b)s(e)f(a)f(comm)m (utativ)m(e)j(ring.)71 b(Let)43 b Fv(X)51 b Fq(2)44 b Fx(Set)e(b)s(e)g(a)g(set.)72 b Fq(F)53 b Fx(:)43 b Fn(K)p Fx(-cAlg)h Fq(\000)-60 b(!)43 b Fx(Set)q(,)h Fq(F)10 b Fx(\()p Fv(A)p Fx(\))43 b(:=)0 5530 y(Map\()p Fv(X)r(;)17 b(A)p Fx(\))38 b(is)g(a)f(co)m(v)-5 b(arian)m(t)38 b(functor.)59 b(A)38 b(represen)m(ting)i(ob)5 b(ject)38 b(for)f Fq(F)47 b Fx(is)38 b(giv)m(en)h(b)m(y)f(the)h(p)s(olynomial)0 5646 y(ring)g(\()p Fn(K)p Fx([)p Fv(X)8 b Fx(])p Fv(;)17 b(\023)39 b Fx(:)f Fv(X)47 b Fq(\000)-60 b(!)38 b Fn(K)p Fx([)p Fv(X)8 b Fx(]\))39 b(with)h(the)f(prop)s(ert)m(y)-8 b(,)41 b(that)e(for)f(all)h(\()p Fv(A;)17 b(f)49 b Fx(:)38 b Fv(X)47 b Fq(\000)-60 b(!)38 b Fv(A)p Fx(\))h(there)h(exists)g(a)p eop end %%Page: 48 48 TeXDict begin 48 47 bop 0 -170 a Fu(48)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fx(unique)34 b Fv(g)d Fq(2)d Fx(Mor)665 44 y Fk(cAlg)809 29 y Fx(\()p Fn(K)p Fx([)p Fv(X)8 b Fx(])p Fv(;)17 b(A)p Fx(\))33 b(suc)m(h)h(that)f Fq(F)10 b Fx(\()p Fv(g)t Fx(\)\()p Fv(\023)p Fx(\))26 b(=)i(Map\()p Fv(X)r(;)17 b(g)t Fx(\)\()p Fv(x)p Fx(\))27 b(=)h Fv(g)t(\023)f Fx(=)h Fv(f)1629 181 y(X)340 b Fn(K)p Fx([)p Fv(X)8 b Fx(])p 1746 158 275 4 v 1938 156 a Fj(-)1871 137 y Fr(\023)1815 416 y(f)1741 307 y Fj(@)1824 390 y(@)1907 473 y(@)1990 556 y(@)2009 575 y(@)-83 b(R)2110 678 y Fv(A:)p 2159 575 4 351 v 2161 575 a Fj(?)2200 408 y Fr(g)0 800 y Fx(\(8\))43 b(Let)g Fn(K)h Fx(b)s(e)g(a)f(comm)m(utativ)m(e)i (ring.)76 b(Let)43 b Fv(X)54 b Fq(2)47 b Fx(Set)c(b)s(e)h(a)f(set.)76 b Fq(F)56 b Fx(:)46 b Fn(K)p Fx(-Alg)g Fq(\000)-60 b(!)46 b Fx(Set,)h Fq(F)10 b Fx(\()p Fv(A)p Fx(\))45 b(:=)0 917 y(Map\()p Fv(X)r(;)17 b(A)p Fx(\))31 b(is)f(a)g(co)m(v)-5 b(arian)m(t)31 b(functor.)43 b(A)30 b(represen)m(ting)j(ob)5 b(ject)31 b(for)f Fq(F)39 b Fx(is)31 b(giv)m(en)h(b)m(y)f(the)f (noncomm)m(uta-)0 1033 y(tiv)m(e)k(p)s(olynomial)g(ring)f(\()p Fn(K)p Fq(h)p Fv(X)8 b Fq(i)p Fv(;)17 b(\023)29 b Fx(:)g Fv(X)37 b Fq(\000)-60 b(!)29 b Fn(K)p Fq(h)p Fv(X)8 b Fq(i)p Fx(\))32 b(with)i(the)g(prop)s(ert)m(y)-8 b(,)34 b(that)f(for)g(all)g(\()p Fv(A;)17 b(f)39 b Fx(:)29 b Fv(X)37 b Fq(\000)-60 b(!)29 b Fv(A)p Fx(\))0 1149 y(there)k(exists)i (a)d(unique)i Fv(g)d Fq(2)d Fx(Mor)1265 1164 y Fk(Alg)1378 1149 y Fx(\()p Fn(K)p Fq(h)p Fv(X)8 b Fq(i)p Fv(;)17 b(A)p Fx(\))32 b(suc)m(h)i(that)e Fq(F)10 b Fx(\()p Fv(g)t Fx(\)\()p Fv(\023)p Fx(\))27 b(=)h(Map\()p Fv(X)r(;)17 b(g)t Fx(\)\()p Fv(x)p Fx(\))27 b(=)h Fv(g)t(\023)f Fx(=)h Fv(f)1623 1301 y(X)329 b Fn(K)p Fq(h)p Fv(X)8 b Fq(i)p 1740 1277 264 4 v 1921 1275 a Fj(-)1860 1257 y Fr(\023)1809 1536 y(f)1736 1427 y Fj(@)1819 1510 y(@)1902 1593 y(@)1985 1676 y(@)2003 1695 y(@)-83 b(R)2105 1797 y Fv(A:)p 2153 1695 4 351 v 2155 1695 a Fj(?)2194 1527 y Fr(g)0 1937 y Ft(Problem)38 b(5.1.)190 b Fx(\(1\))41 b(Giv)m(en)34 b Fv(V)49 b Fq(2)28 b Fn(K)p Fx(-Mo)s(d.)43 b(F)-8 b(or)32 b Fv(A)c Fq(2)g Fn(K)p Fx(-Alg)33 b(de\014ne)632 2094 y Fv(F)14 b Fx(\()p Fv(A)p Fx(\))27 b(:=)h Fq(f)p Fv(f)38 b Fx(:)28 b Fv(V)49 b Fq(\000)-59 b(!)27 b Fv(A)p Fq(j)p Fv(f)43 b Fn(K)p Fx(-linear)p Fv(;)17 b Fq(8)p Fv(v)t(;)g(w)30 b Fq(2)e Fv(V)49 b Fx(:)28 b Fv(f)11 b Fx(\()p Fv(v)t Fx(\))21 b Fq(\001)h Fv(f)11 b Fx(\()p Fv(w)s Fx(\))27 b(=)g(0)p Fq(g)p Fv(:)315 2251 y Fx(Sho)m(w)33 b(that)f(this)i (de\014nes)g(a)e(functor)h Fv(F)41 b Fx(:)28 b Fn(K)p Fx(-Alg)g Fq(\000)-60 b(!)27 b Fx(Set)q(.)148 2367 y(\(2\))42 b(Sho)m(w)25 b(that)f Fv(F)38 b Fx(has)24 b(the)h(algebra)f Fv(D)s Fx(\()p Fv(V)e Fx(\))i(as)g(constructed)i(in)f(Exercise)i(2.1)c (\(3\))h(as)h(a)f(represen)m(ting)315 2483 y(ob)5 b(ject.)0 2657 y Ft(Prop)s(osition)45 b(5.4.)f Fq(F)50 b Fs(has)40 b(a)h(r)-5 b(epr)g(esenting)39 b(obje)-5 b(ct)40 b Fx(\()p Fv(A;)17 b(a)p Fx(\))41 b Fs(if)f(and)g(only)g(if)g(ther)-5 b(e)41 b(is)f(a)g(natur)-5 b(al)41 b(iso-)0 2774 y(morphism)34 b Fv(')27 b Fx(:)h Fq(F)710 2746 y(\030)711 2778 y Fx(=)815 2774 y(Mor)991 2789 y Fp(C)1036 2774 y Fx(\()p Fv(A;)17 b Fq(\000)p Fx(\))35 b(\()p Fs(with)g Fv(a)28 b Fx(=)g Fv(')p Fx(\()p Fv(A)p Fx(\))1987 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b Fs(,)31 b(i.e.)42 b(for)30 b(e)-5 b(ach)29 b Fv(g)i Fx(:)d Fv(X)35 b Fq(\000)-57 b(!)27 b Fv(Y)51 b Fs(ther)-5 b(e)30 b(is)g(a)f(unique)p eop end %%Page: 49 49 TeXDict begin 49 48 bop 992 -170 a Fu(Represen)n(table)26 b(and)f(Adjoin)n(t)g(F)-6 b(unctors,)26 b(the)f(Y)-6 b(oneda)25 b(Lemma)916 b(49)0 29 y Fs(morphism)41 b Fv(A)535 44 y Fr(g)615 29 y Fx(:)g Fv(A)756 44 y Fr(X)864 29 y Fq(\000)-57 b(!)41 b Fv(A)1098 44 y Fr(Y)1200 29 y Fx(\()p Fs(with)h Fq(F)1529 44 y Fr(X)1596 29 y Fx(\()p Fv(A)1707 44 y Fr(g)1747 29 y Fx(\)\()p Fv(a)1874 44 y Fr(X)1942 29 y Fx(\))e(=)h Fq(F)2209 44 y Fr(g)2248 29 y Fx(\()p Fv(A)2359 44 y Fr(Y)2420 29 y Fx(\)\()p Fv(a)2547 44 y Fr(Y)2608 29 y Fx(\)\))h Fs(and)f(the)h(fol)5 b(lowing)40 b(identities)0 146 y(hold)34 b Fv(A)282 161 y Fk(1)317 172 y Fi(X)407 146 y Fx(=)28 b(1)560 161 y Fr(A)613 172 y Fi(X)675 146 y Fv(;)17 b(A)792 161 y Fr(hg)900 146 y Fx(=)27 b Fv(A)1076 161 y Fr(h)1121 146 y Fv(A)1194 161 y Fr(g)1235 146 y Fs(.)44 b(So)35 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Fx(\()p Fv(C)d Fx(\))39 b(b)s(e)g(the)h(uniquely)0 4172 y(determined)34 b(morphism)g(\(b)m(y)g(the)f(Y)-8 b(oneda)32 b(Lemma\))i(in)e Fq(D)j Fx(suc)m(h)g(that)d(the)h(diagram) 1302 4859 y Fq(G)6 b Fx(\()p Fv(C)1482 4823 y Fp(0)1506 4859 y Fv(;)17 b Fx(-)o(\))306 b(Mor)2102 4874 y Fp(D)2163 4859 y Fx(\()p Fq(F)10 b Fx(\()p Fv(C)2398 4823 y Fp(0)2420 4859 y Fx(\))p Fv(;)17 b Fx(-\))p 1648 4837 241 4 v 1806 4835 a Fj(-)1716 4900 y Fr(\030)1747 4919 y Fi(C)1794 4905 y Fh(0)1317 4372 y Fq(G)6 b Fx(\()p Fv(C)r(;)17 b Fx(-)o(\))332 b(Mor)2114 4387 y Fp(D)2175 4372 y Fx(\()p Fq(F)10 b Fx(\()p Fv(C)d Fx(\))p Fv(;)17 b Fx(-)n(\))p 1635 4349 267 4 v 1819 4347 a Fj(-)1726 4315 y Fr(\030)1757 4326 y Fi(C)p 1459 4766 4 351 v 1461 4766 a Fj(?)1232 4607 y Fp(G)t Fk(\()p Fr(f)s(;)p Fx(-)p Fk(\))p 2239 4766 V 2241 4766 a Fj(?)2280 4607 y Fk(Mor)o(\()p Fp(F)7 b Fk(\()p Fr(f)g Fk(\))p Fr(;)p Fx(-)q Fk(\))0 5065 y Fx(comm)m(utes.)71 b(Because)42 b(of)f(the)g(uniqueness)j(of)d Fq(F)10 b Fx(\()p Fv(f)h Fx(\))40 b(and)h(b)s(ecause)h(of)f(the)g (functorialit)m(y)h(of)e Fq(G)47 b Fx(it)41 b(is)0 5182 y(easy)35 b(to)f(see)h(that)f Fq(F)10 b Fx(\()p Fv(f)h(g)t Fx(\))29 b(=)h Fq(F)10 b Fx(\()p Fv(g)t Fx(\))p Fq(F)g Fx(\()p Fv(f)h Fx(\))31 b(and)k Fq(F)10 b Fx(\(1)1930 5197 y Fr(C)1988 5182 y Fx(\))30 b(=)g(1)2211 5197 y Fp(F)7 b Fk(\()p Fr(C)e Fk(\))2416 5182 y Fx(hold)34 b(and)h(that)e Fq(F)44 b Fx(is)35 b(a)e(con)m(tra)m(v)-5 b(arian)m(t)0 5298 y(functor.)0 5414 y(If)36 b Fq(F)183 5378 y Fp(0)239 5414 y Fx(:)d Fq(C)40 b(\000)-60 b(!)33 b(D)38 b Fx(is)f(giv)m(en)g(with)f Fq(G)1340 5386 y(\030)1341 5418 y Fx(=)1450 5414 y(Mor)1627 5429 y Fp(D)1687 5414 y Fx(\()p Fq(F)1807 5378 y Fp(0)1830 5414 y Fx(-)o Fv(;)17 b Fx(-\))35 b(then)i Fv(\036)c Fx(:)g(Mor)2565 5429 y Fp(D)2626 5414 y Fx(\()p Fq(F)10 b Fx(-)o Fv(;)17 b Fx(-)o(\))2925 5386 y Fq(\030)2926 5418 y Fx(=)3036 5414 y(Mor)3212 5429 y Fp(D)3273 5414 y Fx(\()p Fq(F)3393 5378 y Fp(0)3416 5414 y Fx(-)o Fv(;)g Fx(-)o(\).)54 b(Hence)0 5530 y(b)m(y)36 b(the)f(Y)-8 b(oneda)35 b(Lemma)h Fv( )t Fx(\()p Fv(C)7 b Fx(\))31 b(:)h Fq(F)10 b Fx(\()p Fv(C)d Fx(\))1584 5503 y Fq(\030)1585 5534 y Fx(=)1693 5530 y Fq(F)1775 5494 y Fp(0)1798 5530 y Fx(\()p Fv(C)g Fx(\))35 b(is)g(an)g (isomorphism)i(for)d(all)h Fv(C)j Fq(2)33 b(C)6 b Fx(.)50 b(With)36 b(these)0 5646 y(isomorphisms)f(induced)f(b)m(y)f Fv(\036)f Fx(the)h(diagram)p eop end %%Page: 51 51 TeXDict begin 51 50 bop 992 -170 a Fu(Represen)n(table)26 b(and)f(Adjoin)n(t)g(F)-6 b(unctors,)26 b(the)f(Y)-6 b(oneda)25 b(Lemma)916 b(51)846 640 y Fx(Mor)1022 655 y Fp(D)1083 640 y Fx(\()p Fq(F)1203 604 y Fp(0)1226 640 y Fx(\()p Fv(C)1341 604 y Fp(0)1364 640 y Fx(\))p Fv(;)17 b Fx(-)o(\))910 b(Mor)2602 655 y Fp(D)2663 640 y Fx(\()p Fq(F)10 b Fx(\()p Fv(C)2898 604 y Fp(0)2921 640 y Fx(\))p Fv(;)17 b Fx(-)o(\))p 1545 617 845 4 v 2307 615 a Fj(-)1759 685 y Fk(Mor\()p Fr( )r Fk(\()p Fr(C)2044 662 y Fh(0)2068 685 y Fk(\))p Fr(;)p Fx(-)o Fk(\))858 153 y Fx(Mor)1034 168 y Fp(D)1095 153 y Fx(\()p Fq(F)1215 117 y Fp(0)1238 153 y Fx(\()p Fv(C)7 b Fx(\))p Fv(;)17 b Fx(-)o(\))933 b(Mor)2614 168 y Fp(D)2675 153 y Fx(\()p Fq(F)10 b Fx(\()p Fv(C)d Fx(\))p Fv(;)17 b Fx(-)n(\))p 1533 130 868 4 v 2318 128 a Fj(-)1771 92 y Fk(Mor)o(\()p Fr( )r Fk(\()p Fr(C)5 b Fk(\))p Fr(;)p Fx(-)q Fk(\))p 1180 547 4 351 v 1181 547 a Fj(?)732 389 y Fk(Mor\()p Fp(F)944 365 y Fh(0)967 389 y Fk(\()p Fr(f)i Fk(\))p Fr(;)p Fx(-)p Fk(\))p 2740 547 V 2741 547 a Fj(?)2780 388 y Fk(Mor\()p Fp(F)g Fk(\()p Fr(f)g Fk(\))p Fr(;)p Fx(-)p Fk(\))0 803 y Fx(comm)m(utes.)45 b(Hence)34 b(the)f(diagram)1552 1474 y Fq(F)10 b Fx(\()p Fv(C)d Fx(\))338 b Fq(F)2207 1438 y Fp(0)2230 1474 y Fx(\()p Fv(C)7 b Fx(\))p 1815 1451 281 4 v 2013 1449 a Fj(-)1876 1519 y Fr( )r Fk(\()p Fr(C)e Fk(\))1540 987 y Fq(F)10 b Fx(\()p Fv(C)1737 951 y Fp(0)1760 987 y Fx(\))315 b Fq(F)2195 951 y Fp(0)2218 987 y Fx(\()p Fv(C)2333 951 y Fp(0)2356 987 y Fx(\))p 1827 963 258 4 v 2002 961 a Fj(-)1865 926 y Fr( )r Fk(\()p Fr(C)1995 903 y Fh(0)2018 926 y Fk(\))p 1667 1381 4 351 v 1669 1381 a Fj(?)1454 1223 y Fp(F)1511 1199 y Fh(0)1534 1223 y Fk(\()p Fr(f)7 b Fk(\))p 2252 1381 V 2254 1381 a Fj(?)2293 1222 y Fp(F)g Fk(\()p Fr(f)g Fk(\))0 1661 y Fx(comm)m(utes.)45 b(Th)m(us)35 b Fv( )c Fx(:)d Fq(F)37 b(\000)-59 b(!)27 b(F)1229 1625 y Fp(0)1284 1661 y Fx(is)33 b(a)g(natural)f(isomorphism.)1455 b Fo(\003)0 1845 y Fx(5.3.)49 b Ft(Adjoin)m(t)37 b(functors.)0 2021 y(De\014nition)k(5.12.)i Fx(Let)35 b Fq(C)41 b Fx(and)34 b Fq(D)k Fx(b)s(e)d(categories)g(and)g Fq(F)41 b Fx(:)32 b Fq(C)37 b(\000)-59 b(!)31 b(D)37 b Fx(and)e Fq(G)j Fx(:)31 b Fq(D)j(\000)-59 b(!)31 b(C)41 b Fx(b)s(e)35 b(co)m(v)-5 b(arian)m(t)0 2138 y(functors.)41 b Fq(F)31 b Fx(is)22 b(called)h Fs(left)i(adjoint)c Fx(to)g Fq(G)28 b Fx(and)22 b Fq(G)28 b Fs(right)d(adjoint)c Fx(to)h Fq(F)31 b Fx(if)21 b(there)i(is)f(a)g(natural)f(isomorphism)0 2254 y(of)32 b(bifunctors)i Fv(\036)27 b Fx(:)h(Mor)886 2269 y Fp(D)947 2254 y Fx(\()p Fq(F)10 b Fx(-)o Fv(;)17 b Fx(-)o(\))27 b Fq(\000)-59 b(!)27 b Fx(Mor)1561 2269 y Fp(C)1606 2254 y Fx(\(-)p Fv(;)17 b Fq(G)6 b Fx(-)o(\))33 b(from)f Fq(C)2177 2218 y Fr(op)2273 2254 y Fq(\002)23 b(D)35 b Fx(to)d(Set)q(.)0 2431 y Ft(Lemma)38 b(5.13.)j Fs(If)33 b Fq(F)k Fx(:)28 b Fq(C)34 b(\000)-57 b(!)27 b(D)36 b Fs(is)e(left)g(adjoint)f(to)h Fq(G)g Fx(:)27 b Fq(D)k(\000)-57 b(!)27 b(C)40 b Fs(then)33 b Fq(F)44 b Fs(is)33 b(uniquely)h(determine)-5 b(d)33 b(by)0 2547 y Fq(G)41 b Fs(up)35 b(to)g(isomorphism.)43 b(Similarly)34 b Fq(G)41 b Fs(is)35 b(uniquely)g(determine)-5 b(d)34 b(by)h Fq(F)44 b Fs(up)35 b(to)g(isomorphism.)0 2724 y(Pr)-5 b(o)g(of.)41 b Fx(W)-8 b(e)44 b(only)f(pro)m(v)m(e)i(the)f (\014rst)f(claim.)76 b(Assume)45 b(that)e(also)h Fq(F)2578 2688 y Fp(0)2644 2724 y Fx(is)f(left)h(adjoin)m(t)f(to)g Fq(G)49 b Fx(with)44 b Fv(\036)3804 2688 y Fp(0)3873 2724 y Fx(:)0 2841 y(Mor)176 2856 y Fp(D)237 2841 y Fx(\()p Fq(F)357 2805 y Fp(0)380 2841 y Fx(-)o Fv(;)17 b Fx(-)o(\))52 b Fq(\000)-59 b(!)51 b Fx(Mor)923 2856 y Fp(C)968 2841 y Fx(\(-)p Fv(;)17 b Fq(G)6 b Fx(-)o(\).)86 b(Then)48 b(w)m(e)g(ha)m(v)m(e)g(a)f(natural)g(isomorphism)h Fv(\036)3091 2805 y Fp(0)3114 2794 y(\000)p Fk(1)3208 2841 y Fv(\036)k Fx(:)g(Mor)3573 2856 y Fp(D)3634 2841 y Fx(\()p Fq(F)10 b Fx(-)o Fv(;)17 b Fx(-)o(\))0 2957 y Fq(\000)-60 b(!)28 b Fx(Mor)321 2972 y Fp(D)382 2957 y Fx(\()p Fq(F)502 2921 y Fp(0)524 2957 y Fx(-)p Fv(;)17 b Fx(-)o(\).)43 b(By)33 b(Prop)s(osition)g(5.11)f(w)m(e)i(get)e Fq(F)2041 2930 y(\030)2041 2961 y Fx(=)2146 2957 y Fq(F)2228 2921 y Fp(0)2251 2957 y Fx(.)1545 b Fo(\003)0 3134 y Ft(Lemma)41 b(5.14.)i Fs(A)37 b(functor)g Fq(G)g Fx(:)31 b Fq(D)i(\000)-57 b(!)31 b(C)43 b Fs(has)36 b(a)g(left)h(adjoint)f(functor)h(i\013)f(al)5 b(l)36 b(functors)h Fx(Mor)3564 3149 y Fp(C)3609 3134 y Fx(\()p Fv(C)r(;)17 b Fq(G)6 b Fs(-)o Fx(\))0 3250 y Fs(ar)-5 b(e)35 b(r)-5 b(epr)g(esentable.)0 3427 y(Pr)g(o)g(of.)41 b Fx(follo)m(ws)33 b(from)g(5.11.)2773 b Fo(\003)0 3604 y Ft(Lemma)39 b(5.15.)j Fs(L)-5 b(et)35 b Fq(F)i Fx(:)28 b Fq(C)34 b(\000)-57 b(!)27 b(D)38 b Fs(and)c Fq(G)g Fx(:)28 b Fq(D)i(\000)-57 b(!)27 b(C)41 b Fs(b)-5 b(e)35 b(c)-5 b(ovariant)34 b(functors.)45 b(Then)701 3765 y Fx(Nat\(Id)988 3780 y Fp(C)1033 3765 y Fv(;)17 b Fq(G)6 b(F)k Fx(\))27 b Fq(3)i Fx(\010)f Fq(7!)f(G)6 b Fs(-)p Fx(\010)p Fs(-)28 b Fq(2)g Fx(Nat\(Mor)2310 3780 y Fp(D)2371 3765 y Fx(\()p Fq(F)10 b Fs(-)o Fv(;)17 b Fs(-)p Fx(\))p Fv(;)g Fx(Mor)2861 3780 y Fp(C)2906 3765 y Fx(\()p Fs(-)p Fv(;)g Fq(G)6 b Fs(-)p Fx(\)\))0 3926 y Fs(is)35 b(a)f(bije)-5 b(ctive)34 b(map)h(with)f(inverse)g(map)522 4088 y Fx(Nat\(Mor)896 4103 y Fp(D)957 4088 y Fx(\()p Fq(F)10 b Fs(-)o Fv(;)17 b Fs(-)p Fx(\))p Fv(;)g Fx(Mor)1447 4103 y Fp(C)1492 4088 y Fx(\()p Fs(-)p Fv(;)g Fq(G)6 b Fs(-)p Fx(\)\))27 b Fq(3)i Fv(\036)e Fq(7!)g Fv(\036)p Fx(\()p Fs(-)p Fv(;)17 b Fq(F)10 b Fs(-)o Fx(\)\(1)2535 4103 y Fp(F)d Fs(-)2631 4088 y Fx(\))27 b Fq(2)h Fx(Nat\(Id)3077 4103 y Fp(C)3122 4088 y Fv(;)17 b Fq(G)6 b(F)k Fx(\))p Fv(:)0 4249 y Fs(F)-7 b(urthermor)i(e)688 4410 y Fx(Nat)o(\()p Fq(F)10 b(G)c Fv(;)17 b Fx(Id)1165 4425 y Fp(C)1210 4410 y Fx(\))28 b Fq(3)g Fx(\011)g Fq(7!)f Fx(\011)p Fs(-)p Fq(F)10 b Fs(-)27 b Fq(2)h Fx(Nat\(Mor)2324 4425 y Fp(C)2369 4410 y Fx(\()p Fs(-)o Fv(;)17 b Fq(G)6 b Fs(-)p Fx(\))p Fv(;)17 b Fx(Mor)2843 4425 y Fp(D)2904 4410 y Fx(\()p Fq(F)10 b Fs(-)o Fv(;)17 b Fs(-)o Fx(\)\))0 4571 y Fs(is)35 b(a)f(bije)-5 b(ctive)34 b(map)h(with)f(inverse)g(map)527 4733 y Fx(Nat)o(\(Mor)901 4748 y Fp(C)946 4733 y Fx(\()p Fs(-)o Fv(;)17 b Fq(G)6 b Fs(-)p Fx(\))p Fv(;)17 b Fx(Mor)1420 4748 y Fp(D)1481 4733 y Fx(\()p Fq(F)10 b Fs(-)o Fv(;)17 b Fs(-)o Fx(\)\))28 b Fq(3)g Fv( )k Fq(7!)27 b Fv( )t Fx(\()p Fq(G)6 b Fs(-)p Fv(;)17 b Fs(-)p Fx(\)\(1)2542 4748 y Fp(G)t Fs(-)2626 4733 y Fx(\))28 b Fq(2)g Fx(Nat\()p Fq(F)10 b(G)c Fv(;)17 b Fx(Id)3263 4748 y Fp(C)3308 4733 y Fx(\))p Fv(:)0 4910 y Fs(Pr)-5 b(o)g(of.)41 b Fx(The)h(natural)g(transformation)f Fq(G)6 b Fx(-\010-)41 b(is)h(de\014ned)h(as)e(follo)m(ws.)71 b(Giv)m(en)42 b Fv(C)50 b 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Fq(F)10 b Fx(\()p Fv(C)d Fx(\)\)\(1)1491 5544 y Fp(F)g Fk(\()p Fr(C)e Fk(\))1659 5529 y Fx(\))28 b(=)f(Mor)2005 5544 y Fp(C)2050 5529 y Fx(\()p Fv(C)r(;)17 b Fq(G)6 b Fx(\()p Fv(f)11 b Fx(\)\))p Fv(\036)p Fx(\()p Fv(C)r(;)17 b Fq(F)10 b Fx(\()p Fv(C)d Fx(\)\)\(1)3014 5544 y Fp(F)g Fk(\()p Fr(C)e Fk(\))3181 5529 y Fx(\))681 5645 y(=)27 b Fv(\036)p Fx(\()p Fv(C)r(;)17 b(D)s Fx(\))g(Mor)1309 5660 y Fp(D)1370 5645 y Fx(\()p Fq(F)10 b Fx(\()p Fv(C)d Fx(\))p Fv(;)17 b(f)11 b Fx(\)\(1)1871 5660 y Fp(F)c Fk(\()p Fr(C)e Fk(\))2041 5645 y Fx(\))27 b(=)h Fv(\036)p Fx(\()p Fv(C)r(;)17 b(D)s Fx(\)\()p Fv(f)11 b Fx(\))p Fv(:)p eop end %%Page: 52 52 TeXDict begin 52 51 bop 0 -170 a Fu(52)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fx(So)32 b(the)h(t)m(w)m(o)h(maps)f (are)g(in)m(v)m(erses)i(of)d(eac)m(h)i(other.)0 146 y(The)g(second)f (part)g(of)f(the)h(lemma)g(is)g(pro)m(v)m(ed)h(similarly)-8 b(.)1704 b Fo(\003)0 306 y Ft(Prop)s(osition)38 b(5.16.)j Fs(L)-5 b(et)435 463 y Fv(\036)28 b 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5298 y Fs(as)34 b(bimo)-5 b(dules.)148 5414 y Fx(\(4\))42 b Fv(A)415 5386 y Fq(\030)416 5418 y Fx(=)521 5414 y(Hom)724 5429 y Fr(B)784 5414 y Fx(\()p Fv(:Q;)17 b(:Q)p Fx(\))1140 5386 y Fq(\030)1141 5418 y Fx(=)1245 5414 y(Hom)1449 5429 y Fr(B)1509 5414 y Fx(\()p Fv(P)s(:;)g(P)s(:)p Fx(\))315 5530 y Fv(B)421 5503 y Fq(\030)422 5534 y Fx(=)527 5530 y(Hom)730 5545 y Fr(A)787 5530 y Fx(\()p Fv(:P)s(;)g(:P)d Fx(\))1130 5503 y Fq(\030)1131 5534 y Fx(=)1235 5530 y(Hom)1439 5545 y Fr(A)1496 5530 y Fx(\()p Fv(Q:;)j(Q:)p Fx(\))315 5646 y Fs(as)34 b Fn(K)p Fs(-algebr)-5 b(as)34 b(and)h(as)f(bimo)-5 b(dules.)p eop end %%Page: 61 61 TeXDict begin 61 60 bop 1579 -170 a Fu(The)26 b(Morita)h(Theorems)1504 b(61)148 29 y Fx(\(5\))42 b Fv(P)d Fq(\012)494 44 y Fr(B)582 29 y Fs(-)f Fx(:)h Fv(B)5 b Fs(-)p Fx(Mo)s(d)39 b Fq(\000)-57 b(!)38 b Fv(A)p Fs(-)p Fx(Mo)s(d)j Fs(and)f Fv(Q)27 b Fq(\012)1947 44 y Fr(A)2030 29 y Fs(-)39 b Fx(:)g Fv(A)p Fs(-)p Fx(Mo)s(d)f Fq(\000)-57 b(!)38 b Fv(B)5 b Fs(-)p 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Fx(\(1\))i(The)g(isomorphisms)i (from)e(Theorem)h(1.22)e(\(5\))g(map)h Fv(g)49 b Fq(2)c Fx(Hom)2978 1792 y Fr(B)s Fx(-)p Fr(B)3128 1777 y Fx(\()p Fv(:Q)29 b Fq(\012)3376 1792 y Fr(A)3463 1777 y Fv(P)s(:;)17 b(:B)5 b(:)p Fx(\))42 b(to)0 1893 y(homomorphisms)e(of)d(bimo)s(dules)i Fv(g)1365 1908 y Fk(1)1441 1893 y Fx(:)d Fv(P)50 b Fq(\000)-59 b(!)36 b Fx(Hom)1974 1908 y Fr(B)2035 1893 y Fx(\()p Fv(:Q;)17 b(:B)5 b Fx(\))38 b(and)g Fv(g)2645 1908 y Fk(2)2720 1893 y Fx(:)f Fv(Q)g Fq(\000)-60 b(!)36 b Fx(Hom)3255 1908 y Fr(B)3315 1893 y Fx(\()p Fv(P)s(:;)17 b(B)5 b(:)p Fx(\).)59 b(F)-8 b(ur-)0 2009 y(thermore)48 b Fv(f)58 b Fx(induces)49 b(homomorphisms)g(of)e(bimo)s(dules)i Fv(f)2295 2024 y Fk(1)2387 2009 y Fx(:)k Fv(P)66 b Fq(\000)-59 b(!)52 b Fx(Hom)2970 2024 y Fr(A)3027 2009 y Fx(\()p Fv(Q:;)17 b(A:)p Fx(\))47 b(and)g Fv(f)3650 2024 y Fk(2)3743 2009 y Fx(:)53 b Fv(Q)0 2125 y Fq(\000)-60 b(!)28 b Fx(Hom)348 2140 y Fr(A)405 2125 y Fx(\()p Fv(:P)s(;)17 b(:A)p Fx(\).)0 2241 y(If)38 b Fv(g)j 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b(let)49 b Fv(')55 b Fq(2)0 727 y Fx(End)175 742 y Fk(funkt)344 727 y Fx(\(Id)471 742 y Fr(A)p Fx(-)11 b Fk(Mo)r(d)713 727 y Fx(\))28 b(=:)g Fv(E)6 b Fx(\()p Fv(A)p Fx(\).)43 b(W)-8 b(e)33 b(de\014ne)h Fv(')1721 691 y Fp(0)1772 727 y Fq(2)28 b Fv(E)6 b Fx(\()p Fv(B)f Fx(\))32 b(b)m(y)1274 1449 y Fv(S)6 b(T)14 b Fx(\()1449 1464 y Fr(B)1509 1449 y Fv(M)c Fx(\))598 b Fv(S)6 b(T)14 b Fx(\()2424 1464 y Fr(B)2484 1449 y Fv(M)c Fx(\))p 1680 1427 541 4 v 2138 1425 a Fj(-)1759 1519 y Fv(S)c('T)14 b Fx(\()p Fv(M)c Fx(\))1380 979 y Fr(B)1440 964 y Fv(M)2355 979 y Fr(B)2415 964 y Fv(M)p 1574 939 753 4 v 2244 937 a Fj(-)1816 893 y Fv(')1880 857 y Fp(0)1903 893 y Fx(\()p Fv(M)g Fx(\))p 1461 1356 4 351 v 1463 1356 a Fj(?)1182 1206 y Fv(\014)c Fx(\()p Fv(M)k Fx(\))p 2436 1356 V 2437 1356 a Fj(?)2476 1206 y Fv(\014)c Fx(\()p Fv(M)k Fx(\))0 1679 y(where)38 b Fv(S)h Fx(:)c Fv(A)p Fx(-)o(Mo)s(d)f Fq(\000)-59 b(!)34 b Fv(B)5 b Fx(-)o(Mo)s(d,)38 b Fv(T)47 b Fx(:)35 b Fv(B)5 b Fx(-)o(Mo)s(d)34 b Fq(\000)-59 b(!)34 b Fv(A)p Fx(-)o(Mo)s(d)j(are)f (the)h(m)m(utually)g(in)m(v)m(erse)i(equiv)-5 b(alences)0 1795 y(from)37 b(\(5\),)g(and)h Fv(\013)d Fx(:)h(Id)868 1810 y Fr(A)p Fx(-)11 b Fk(Mo)r(d)1145 1795 y Fq(\000)-59 b(!)35 b Fv(T)14 b(S)42 b Fx(and)37 b Fv(\014)k Fx(:)35 b(Id)1913 1810 y Fr(B)s Fx(-)11 b Fk(Mo)r(d)2194 1795 y Fq(\000)-60 b(!)35 b Fv(S)6 b(T)50 b Fx(resp.)58 b(are)37 b(the)g(asso)s(ciated)h(isomor-)0 1911 y(phisms.)45 b(Analogously)33 b(w)m(e)h(asso)s(ciate)f(with)h(eac)m(h)f Fv( )f Fq(2)c Fv(E)6 b Fx(\()p Fv(B)f Fx(\))32 b(an)h(elemen)m(t)h Fv( )2930 1875 y Fp(0)2981 1911 y Fq(2)28 b Fv(E)6 b Fx(\()p Fv(A)p Fx(\))33 b(b)m(y)1270 2634 y Fv(T)14 b(S)6 b Fx(\()1445 2649 y Fr(A)1501 2634 y Fv(N)k Fx(\))618 b Fv(T)14 b(S)6 b Fx(\()2420 2649 y Fr(A)2476 2634 y Fv(N)k Fx(\))p 1657 2611 560 4 v 2134 2609 a Fj(-)1752 2703 y Fv(T)k( )t(S)6 b Fx(\()p Fv(N)k Fx(\))1376 2163 y Fr(A)1433 2148 y Fv(N)2351 2163 y Fr(A)2408 2148 y Fv(N)p 1550 2123 773 4 v 2240 2121 a Fj(-)1809 2077 y Fv( )1876 2041 y Fp(0)1899 2077 y Fx(\()p Fv(N)g Fx(\))p 1447 2541 4 351 v 1449 2541 a Fj(?)1183 2390 y Fv(\013)q Fx(\()p Fv(N)g Fx(\))p 2422 2541 V 2424 2541 a Fj(?)2463 2390 y Fv(\013)q Fx(\()p Fv(N)g Fx(\))2689 2609 y Fv(:)0 2868 y Fx(The)34 b(comp)s(ositions)h(of)e Fv( )g Fq(7!)c Fv( )1190 2831 y Fp(0)1246 2868 y Fx(and)34 b Fv(')29 b Fq(7!)g Fv(')1723 2831 y Fp(0)1779 2868 y Fx(in)34 b(eac)m(h)g(direction)h(de\014ne)f(isomorphisms,)i(hence)f(eac)m(h)0 2984 y(single)41 b(map)f(is)g(an)g(isomorphism.)67 b(One)40 b(of)f(the)i(t)m(w)m(o)f(comp)s(ositions)h(is)f(con)m(tained)h(in)f (the)h(follo)m(wing)0 3100 y(diagram.)1296 3298 y Fv(N)1141 b(N)p 1414 3266 1073 4 v 2404 3264 a Fj(-)1815 3220 y Fv(')1879 3184 y Fp(00)1921 3220 y Fx(\()p Fv(N)10 b Fx(\))1190 3679 y Fv(T)k(S)6 b Fx(\()p Fv(N)k Fx(\))918 b Fv(T)14 b(S)6 b Fx(\()p Fv(N)k Fx(\))p 1520 3656 860 4 v 2297 3654 a Fj(-)1756 3610 y Fv(T)k(')1891 3574 y Fp(0)1914 3610 y Fv(S)6 b Fx(\()p Fv(N)k Fx(\))1084 4069 y Fv(T)k Fx(\()p Fv(S)6 b(T)14 b Fx(\))p Fv(S)6 b Fx(\()p Fv(N)k Fx(\))705 b Fv(T)14 b Fx(\()p Fv(S)6 b(T)14 b Fx(\))p Fv(S)6 b Fx(\()p Fv(N)k Fx(\))p 1626 4046 648 4 v 2191 4044 a Fj(-)1699 4000 y Fv(T)k(S)6 b('T)14 b(S)6 b Fx(\()p Fv(N)k Fx(\))1190 4459 y Fv(T)k(S)6 b Fx(\()p Fv(N)k Fx(\))918 b Fv(T)14 b(S)6 b Fx(\()p Fv(N)k Fx(\))p 1520 4436 860 4 v 2297 4434 a Fj(-)1768 4390 y Fv(T)k(S)6 b(')p Fx(\()p Fv(N)k Fx(\))1296 4858 y Fv(N)1141 b(N)p 1414 4826 1073 4 v 2404 4824 a Fj(-)1836 4780 y Fv(')p Fx(\()p Fv(N)10 b Fx(\))p 1339 3586 4 254 v 1341 3586 a Fj(?)1075 3484 y Fv(\013)q Fx(\()p Fv(N)g Fx(\))p 2558 3586 V 2559 3586 a Fj(?)2598 3484 y Fv(\013)q Fx(\()p Fv(N)g Fx(\))p 1339 3976 V 1341 3976 a Fj(?)940 3874 y Fv(T)k(\014)6 b(S)g Fx(\()p Fv(N)k Fx(\))p 2558 3976 V 2559 3976 a Fj(?)2598 3874 y Fv(T)k(\014)6 b(S)g Fx(\()p Fv(N)k Fx(\))p 1339 4366 V 1341 4195 a Fj(6)938 4264 y Fv(T)k(S)6 b(\013)q Fx(\()p Fv(N)k Fx(\))p 2558 4366 V 2559 4195 a Fj(6)2598 4264 y Fv(T)k(S)6 b(\013)q Fx(\()p Fv(N)k Fx(\))p 1339 4756 V 1341 4585 a Fj(6)1075 4654 y Fv(\013)q Fx(\()p Fv(N)g Fx(\))p 2558 4756 V 2559 4585 a Fj(6)2598 4654 y Fv(\013)q Fx(\()p Fv(N)g Fx(\))0 4978 y(Th)m(us)34 b(the)f(map)g Fv(')27 b Fq(7!)h Fv(')915 4942 y Fp(00)990 4978 y Fx(is)33 b(an)f(inner)i(automorphism)f(of)f Fv(E)6 b Fx(\()p Fv(A)p Fx(\),)33 b(hence)h(it)e(is)h(bijectiv)m(e.)461 b Fo(\003)0 5162 y Ft(Theorem)38 b(7.10.)k Fx(\(Morita)32 b(I)s(I\))0 5278 y Fs(L)-5 b(et)39 b Fv(S)i Fx(:)35 b Fv(A)p Fs(-)p Fx(Mo)s(d)g Fq(\000)-57 b(!)35 b Fv(B)5 b Fs(-)p Fx(Mo)s(d)39 b Fs(and)f Fv(T)49 b Fx(:)35 b Fv(B)5 b Fs(-)p Fx(Mo)s(d)35 b Fq(\000)-57 b(!)34 b Fv(A)p Fs(-)p Fx(Mo)s(d)39 b Fs(b)-5 b(e)39 b(mutual)5 b(ly)39 b(inverse)f Fn(K)p Fs(-e)-5 b(quivalenc)g(es.)0 5395 y(L)g(et)171 5410 y Fr(A)228 5395 y Fv(P)291 5410 y Fr(B)384 5395 y Fx(:=)33 b Fv(T)14 b Fx(\()p Fv(B)5 b Fx(\))37 b Fs(and)975 5410 y Fr(B)1036 5395 y Fv(Q)1113 5410 y Fr(A)1202 5395 y Fx(:=)c Fv(S)6 b Fx(\()p Fv(A)p Fx(\))p Fs(.)53 b(Then)36 b(ther)-5 b(e)38 b(ar)-5 b(e)37 b(isomorphisms)f Fv(f)43 b Fx(:)3074 5410 y Fr(A)3131 5395 y Fv(P)37 b Fq(\012)3308 5410 y Fr(B)3394 5395 y Fv(Q)3471 5410 y Fr(A)3561 5395 y Fq(\000)-57 b(!)3713 5410 y Fr(A)3770 5395 y Fv(A)3843 5410 y Fr(A)0 5511 y Fs(and)34 b Fv(g)d Fx(:)322 5526 y Fr(B)383 5511 y Fv(Q)22 b Fq(\012)559 5526 y Fr(A)639 5511 y Fv(P)702 5526 y Fr(B)790 5511 y Fq(\000)-57 b(!)937 5526 y Fr(B)998 5511 y Fv(B)1072 5526 y Fr(B)1133 5511 y Fs(,)34 b(such)h(that)g Fx(\()p Fv(A;)17 b(B)5 b(;)17 b(P)s(;)g(Q;)g(f)5 b(;)17 b(g)t Fx(\))33 b Fs(is)i(a)f(Morita)i(c)-5 b(ontext.)0 5627 y(F)e(urthermor)i(e)34 b(the)h(fol)5 b(lowing)33 b(hold)i Fv(S)1441 5599 y Fq(\030)1442 5631 y Fx(=)1546 5627 y Fv(Q)23 b Fq(\012)1723 5642 y Fr(A)1802 5627 y Fs(-)35 b(and)f Fv(T)2160 5599 y Fq(\030)2160 5631 y Fx(=)2265 5627 y Fv(P)h Fq(\012)2440 5642 y Fr(B)2524 5627 y Fs(-)p Fv(:)p eop end %%Page: 64 64 TeXDict begin 64 63 bop 0 -170 a Fu(64)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Ft(Theorem)38 b(7.11.)k Fx(\(Morita)32 b(I)s(I)s(I\))0 146 y Fs(L)-5 b(et)46 b Fv(P)61 b Fq(2)49 b Fv(A)p Fs(-)p Fx(Mo)s(d)c Fs(b)-5 b(e)46 b(a)g(\014nitely)f(gener)-5 b(ate)g(d)45 b(pr)-5 b(oje)g(ctive)45 b(gener)-5 b(ator)46 b Fx(\(=)e(progenerator\))p Fs(.)77 b(Then)45 b(the)0 262 y(Morita)c(c)-5 b(ontext)40 b Fx(\()p Fv(A;)17 b Fx(Hom)1028 277 y Fr(A)1085 262 y Fx(\()p Fv(:P)s(;)g(:P)d Fx(\))p Fv(;)j(P)s(;)g(Q;)g(f)47 b Fx(=)38 b(ev)r Fv(;)17 b(g)41 b Fx(=)d(db)q(\))i Fs(is)g(strict,)i (i.e.)62 b Fv(f)51 b Fs(and)40 b Fv(g)k Fs(ar)-5 b(e)40 b(epimor-)0 378 y(phisms.)0 562 y(Pr)-5 b(o)g(of.)41 b Fx(Since)553 577 y Fr(A)610 562 y Fv(P)k Fx(is)33 b(\014nitely)h (generated)f(pro)5 b(jectiv)m(e,)34 b Fv(g)d Fx(=)d(db)k(is)h(an)g (isomorphism)g(\(3.19\).)43 b(Since)3767 577 y Fr(A)3824 562 y Fv(P)0 679 y Fx(is)33 b(a)f(generator,)h Fv(f)38 b Fx(=)28 b(ev)35 b(is)e(an)f(epimorphism)j(\(3.24\).)1773 b Fo(\003)0 866 y Fs(Pr)-5 b(o)g(of)35 b(of)f(7.10:)41 b Fx(1.)52 b(Giv)m(en)36 b Fv(S)6 b Fx(,)36 b Fv(T)14 b Fx(.)51 b(Then)37 b Fv(S)h Fx(:)33 b(Hom)1937 881 y Fr(A)1994 866 y Fx(\()p Fv(:M)5 b(;)17 b(:N)10 b Fx(\))33 b Fq(3)g Fv(f)43 b Fq(7!)32 b Fv(S)6 b Fx(\()p Fv(f)11 b Fx(\))32 b Fq(2)h Fx(Hom)3245 881 y Fr(B)3306 866 y Fx(\()p Fv(:S)6 b(M)f(;)17 b(:S)6 b(N)k Fx(\))35 b(is)0 982 y(an)d(isomorphism.)46 b(Let)32 b Fv(\013)d Fx(:)e Fv(T)14 b(S)1228 954 y Fq(\030)1228 986 y Fx(=)1333 982 y(Id)1422 997 y Fr(A)p Fx(-)d Fk(Mo)r(d)1665 982 y Fx(.)43 b(Then)148 1184 y(Hom)351 1199 y Fr(A)408 1184 y Fx(\()p Fv(:M)5 b(;)17 b(:N)10 b Fx(\))148 b(Hom)1120 1199 y Fr(B)1181 1184 y Fx(\()p Fv(:S)6 b(M)f(;)17 b(:S)6 b(N)k Fx(\))p 798 1161 82 4 v 797 1159 a Fj(-)816 1140 y Fr(S)p 1695 1161 92 4 v 1704 1159 a Fj(-)1715 1140 y Fr(T)1815 1184 y Fx(Hom)2018 1199 y Fr(A)2075 1184 y Fx(\()p Fv(:T)k(S)6 b(M)f(;)17 b(:T)d(S)6 b(N)k Fx(\))420 b(Hom)3333 1199 y Fr(A)3390 1184 y Fx(\()p Fv(:M)5 b(;)17 b(:N)10 b Fx(\))p 2738 1161 355 4 v 3010 1159 a Fj(-)2718 1123 y Fk(Hom\()p Fr(\013)2937 1100 y Fh(\000)p Fg(1)3021 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b(is)h(an)f(isomorphism,)55 b Fv(T)63 b Fx(is)50 b(an)f(epimorphism)i(where)g Fv(T)69 b Fx(:)57 b(Hom)3173 2424 y Fr(B)3234 2409 y Fx(\()p Fv(:S)6 b(M)f(;)17 b(:S)6 b(N)k Fx(\))56 b Fq(\000)-60 b(!)0 2525 y Fx(Hom)203 2540 y Fr(A)260 2525 y Fx(\()p Fv(:T)14 b(S)6 b(M)f(;)17 b(:T)d(S)6 b(N)k Fx(\).)74 b(By)43 b(symmetry)i Fv(T)56 b Fx(is)44 b(a)e(monomorphism.)75 b(Hence)44 b Fv(T)57 b Fx(is)43 b(an)f(isomorphism)0 2641 y(in)33 b(the)g(ab)s(o)m(v)m(e)g(map.)44 b(Th)m(us)34 b Fv(S)k Fx(is)33 b(an)g(isomorphism.)0 2798 y(2.)57 b(Hom)336 2813 y Fr(B)397 2798 y Fx(\()p Fv(:S)6 b(M)f(;)17 b(:N)10 b Fx(\))893 2742 y Fr(T)860 2798 y Fq(\000)-60 b(!)35 b Fx(Hom)1216 2813 y Fr(A)1273 2798 y Fx(\()p Fv(:T)14 b(S)6 b(M)f(;)17 b(:T)d(N)c Fx(\))1877 2734 y Fk(Hom\()p Fr(\013)2096 2711 y Fh(\000)p Fg(1)2179 2734 y Fr(;)p Fk(id\))2001 2798 y Fq(\000)-16 b(!)159 b Fx(Hom)2524 2813 y Fr(A)2581 2798 y Fx(\()p Fv(:M)5 b(;)17 b(:T)d(N)c Fx(\))37 b(is)h(a)f(natural)g(isomor-)0 2915 y(phism.)80 b(It)44 b(is)h(clear)f(that)g(this)h(is)g(an)f (isomorphism.)80 b(Since)45 b Fv(T)58 b Fx(is)45 b(a)e(functor,)48 b(the)c(\014rst)h(map)g(is)f(a)0 3031 y(natural)38 b(transformation.)61 b(The)40 b(second)f(map)g(is)g(a)f(natural)g(transformation,)i(since)g Fv(\013)f Fx(is)g(a)f(natural)0 3147 y(transformation.)43 b(In)33 b(particular,)g Fv(S)38 b Fx(is)c(left)e(adjoin)m(t)h(to)f Fv(T)14 b Fx(.)0 3263 y(3.)87 b Fv(S)6 b Fx(\()p Fq(\010)344 3278 y Fr(i)p Fp(2)p Fr(I)456 3263 y Fv(M)550 3278 y Fr(i)578 3263 y Fx(\))669 3236 y Fq(\030)670 3268 y Fx(=)799 3263 y Fq(\010)876 3278 y Fr(i)p Fp(2)p Fr(I)987 3263 y Fv(S)g Fx(\()p Fv(M)1185 3278 y Fr(i)1214 3263 y Fx(\),)51 b(since)d Fv(S)53 b Fx(is)48 b(a)f(left)h(adjoin)m(t)f(functor)g(and)h (th)m(us)g(preserv)m(es)i(direct)0 3380 y(copro)s(ducts.)0 3496 y(4.)45 b(If)33 b Fv(f)40 b Fq(2)29 b Fv(B)5 b Fx(-Mo)s(d)33 b(is)h(an)f(epimorphism,)i(then)f Fv(T)14 b(f)39 b Fq(2)29 b Fv(A)p Fx(-Mo)s(d)k(is)h(an)f(epimorphism,)i(to)s(o.)45 b(In)33 b(fact,)h(let)0 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Fx(=)c Fv(S)6 b(h)p Fx(,)33 b(hence)h Fv(g)d Fx(=)c Fv(h)p Fx(.)0 4717 y(5.)41 b(If)24 b Fv(P)41 b Fq(2)28 b Fv(A)p Fx(-Mo)s(d)c(is)h(pro)5 b(jectiv)m(e,)29 b(then)c Fv(S)6 b(P)41 b Fq(2)28 b Fv(B)5 b Fx(-)o(Mo)s(d)25 b(is)g(pro)5 b(jectiv)m(e.)43 b(In)25 b(fact)f(giv)m(en)i(an)e(epimorphism)0 4833 y Fv(f)59 b Fx(:)48 b Fv(M)59 b Fq(\000)-59 b(!)48 b Fv(N)55 b Fx(in)45 b Fv(B)5 b Fx(-)o(Mo)s(d)45 b(and)g(a)f(homomorphism)i Fv(g)52 b Fx(:)c Fv(S)6 b(P)61 b Fq(\000)-59 b(!)48 b Fv(N)10 b Fx(.)80 b(Then)46 b Fv(T)14 b(f)58 b Fx(:)49 b Fv(T)14 b(M)58 b Fq(\000)-59 b(!)48 b Fv(T)14 b(N)0 4949 y Fx(is)46 b(an)f(epimorphism)i(and)e Fv(T)14 b(g)52 b Fx(:)d Fv(T)14 b(S)6 b(P)62 b Fq(\000)-60 b(!)49 b Fv(T)14 b(N)55 b Fx(is)46 b(in)f Fv(A)p Fx(-Mo)s(d.)81 b(Since)46 b Fv(\013)j Fx(:)h Fv(T)14 b(S)6 b(P)3294 4921 y Fq(\030)3295 4953 y Fx(=)3421 4949 y Fv(P)14 b Fx(,)48 b(there)d(is)0 5065 y(an)g Fv(h)50 b Fx(:)g Fv(P)63 b Fq(\000)-60 b(!)49 b Fv(T)14 b(M)56 b Fx(with)46 b Fv(T)14 b(f)41 b Fq(\016)31 b Fv(h)50 b Fx(=)f Fv(T)14 b(g)34 b Fq(\016)d Fv(\013)1847 5029 y Fp(\000)p Fk(1)1986 5065 y Fx(or)45 b Fv(T)14 b(f)41 b Fq(\016)31 b Fv(h)g Fq(\016)g Fv(\013)50 b Fx(=)f Fv(T)14 b(g)t Fx(.)81 b(W)-8 b(e)46 b(apply)g Fv(S)51 b Fx(and)46 b(get)0 5182 y Fv(S)6 b(T)14 b(f)34 b Fq(\016)23 b Fv(S)6 b Fx(\()p Fv(h)24 b Fq(\016)f Fv(\013)q Fx(\))31 b(=)h Fv(S)6 b(T)14 b(g)t Fx(,)34 b(where)i Fv(S)6 b Fx(\()p Fv(h)23 b Fq(\016)h Fv(\013)q Fx(\))31 b Fq(2)h Fx(Hom)2012 5197 y Fr(B)2073 5182 y Fx(\()p Fv(:S)6 b(T)14 b(S)6 b(P)s(;)17 b(:S)6 b(T)14 b(M)c Fx(\).)49 b(Since)36 b Fv(\014)h Fx(:)31 b Fv(S)6 b(T)14 b(M)3513 5154 y Fq(\030)3514 5186 y Fx(=)3622 5182 y Fv(M)c Fx(,)36 b(w)m(e)0 5298 y(ha)m(v)m(e)f(an)e(isomorphism)i (Hom\()p Fv(\014)1235 5262 y Fp(\000)p Fk(1)1329 5298 y Fv(;)17 b(\014)6 b Fx(\))29 b(:)g(Hom)1760 5313 y Fr(B)1821 5298 y Fx(\()p Fv(:S)6 b(T)14 b(S)6 b(P)s(;)17 b(:S)6 b(T)14 b(M)c Fx(\))28 b Fq(\000)-60 b(!)29 b Fx(Hom)2882 5313 y Fr(B)2943 5298 y Fx(\()p Fv(:S)6 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b(w)m(e)h(get)0 5646 y Fv(S)6 b(T)14 b(f)32 b Fq(\016)22 b Fv(S)6 b(T)14 b(k)31 b Fx(=)c Fv(S)6 b(T)14 b(g)31 b Fx(=)c Fv(S)6 b(T)14 b Fx(\()p Fv(f)32 b Fq(\016)22 b Fv(k)s Fx(\))33 b(and)g(th)m(us)g Fv(g)e Fx(=)d Fv(f)33 b Fq(\016)22 b Fv(k)s Fx(.)43 b(So)33 b Fv(S)6 b(P)45 b Fx(is)33 b(pro)5 b(jectiv)m(e.)p eop end %%Page: 65 65 TeXDict begin 65 64 bop 1579 -170 a Fu(The)26 b(Morita)h(Theorems)1504 b(65)0 29 y Fx(6.)40 b Fv(S)6 b(A)23 b Fx(is)h(\014nitely)h(generated)f (as)f(a)g Fv(B)5 b Fx(-mo)s(dule:)39 b(Since)25 b Fv(S)6 b(A)23 b Fx(is)h(pro)5 b(jectiv)m(e,)27 b(w)m(e)d(ha)m(v)m(e)h Fv(S)6 b(A)s Fq(\010)s Fv(X)3450 2 y Fq(\030)3451 34 y Fx(=)3555 -45 y Fl(L)3666 59 y Fr(i)p Fp(2)p Fr(I)3794 29 y Fv(B)f Fx(.)0 146 y(By)29 b(\(3\))g(applied)g(to)f Fv(T)43 b Fx(w)m(e)30 b(get)e Fv(A)14 b Fq(\010)g Fv(T)g(X)1520 118 y Fq(\030)1520 150 y Fx(=)1625 146 y Fv(T)g(S)6 b(A)14 b Fq(\010)g Fv(T)g(X)2128 118 y Fq(\030)2128 150 y Fx(=)2233 71 y Fl(L)2344 175 y Fr(i)p Fp(2)p Fr(I)2471 146 y Fv(T)g(B)5 b Fx(.)42 b(Since)30 b Fv(A)f Fx(is)g(\014nitely)h(generated,)0 262 y(the)g(image)f(of)g Fv(A)g Fx(in)763 187 y Fl(L)874 291 y Fr(i)p Fp(2)p Fr(I)1001 262 y Fv(T)14 b(B)34 b Fx(is)c(already)g(a)f(direct)h(summand)g(in)f(a)g(\014nite)h(direct)g (subsum)3465 187 y Fl(L)3576 291 y Fr(i)p Fp(2)p Fr(E)3723 262 y Fv(T)14 b(B)5 b Fx(,)0 378 y(so)38 b Fv(A)26 b Fq(\010)g Fv(Y)442 351 y Fq(\030)443 382 y Fx(=)557 304 y Fl(L)667 407 y Fr(i)p Fp(2)p Fr(E)815 378 y Fv(T)14 b(B)5 b Fx(.)59 b(Hence)39 b Fv(S)6 b(A)26 b Fq(\010)g Fv(S)6 b(Y)1795 351 y Fq(\030)1796 382 y Fx(=)1910 304 y Fl(L)2020 407 y Fr(i)p Fp(2)p Fr(E)2168 378 y Fv(S)g(T)14 b(B)2420 351 y Fq(\030)2421 382 y Fx(=)2534 304 y Fl(L)2645 407 y Fr(i)p Fp(2)p Fr(E)2792 378 y Fv(B)43 b Fx(and)38 b(th)m(us)h Fv(S)6 b(A)38 b Fx(is)h(\014nitely)0 495 y(generated.)0 611 y(7.)56 b(If)37 b Fv(G)e Fq(2)h Fv(A)p Fx(-)o(Mo)s(d)h(is)h(a)e(generator)h(then)h Fv(S)6 b(G)34 b Fq(2)i Fv(B)5 b Fx(-Mo)s(d)37 b(is)g(also)g(a)g(generator.)56 b(In)37 b(fact)g(let)g(\()p Fv(f)46 b Fx(:)35 b Fv(M)0 727 y Fq(\000)-60 b(!)43 b Fv(N)10 b Fx(\))44 b Fq(6)p Fx(=)g(0)d(in)h Fv(B)5 b Fx(-Mo)s(d.)71 b(Then)43 b Fv(T)14 b(f)54 b Fq(6)p Fx(=)43 b(0,)h(hence)g(there)e(is)h(a)e Fv(g)47 b Fx(:)c Fv(G)h Fq(\000)-60 b(!)43 b Fv(T)14 b(M)52 b Fx(with)43 b Fv(T)14 b(f)39 b Fq(\016)28 b Fv(g)47 b Fq(6)p Fx(=)c(0.)0 843 y(Consequen)m(tly)36 b Fv(S)6 b(T)14 b(f)32 b Fq(\016)22 b Fv(S)6 b(g)31 b Fq(6)p Fx(=)c(0)33 b(and)f Fv(f)h Fq(\016)22 b Fx(\()p Fv(\013)h Fq(\016)f Fv(S)6 b(g)t Fx(\))27 b(=)g Fv(\013)c Fq(\016)f Fv(S)6 b(T)14 b(f)32 b Fq(\016)22 b Fv(S)6 b(g)31 b Fq(6)p Fx(=)d(0.)0 959 y(8.)43 b(This)34 b(sho)m(ws)g(that)f Fv(S)6 b Fx(\()p Fv(A)p Fx(\))32 b(is)h(a)f(\014nitely)i(generated)f(pro)5 b(jectiv)m(e)35 b(generator.)0 1076 y(\(Remark:)61 b(An)42 b(equiv)-5 b(alence)43 b Fv(S)k Fx(alw)m(a)m(ys)c(maps)e(\014nitely)i (generated)f(mo)s(dules)g(to)f(\014nitely)h(generated)0 1192 y(mo)s(dules.)j(W)-8 b(e)33 b(will)g(giv)m(e)g(the)g(pro)s(of)f (further)h(do)m(wn)g(in)g(Prop)s(osition)g(7.12.\))0 1337 y(9.)43 b Fv(A)220 1309 y Fq(\030)221 1341 y Fx(=)325 1337 y(Hom)528 1352 y Fr(B)589 1337 y Fx(\()p Fv(:S)6 b(A;)17 b(:S)6 b(A)p Fx(\))32 b(as)h(algebras,)g(since)h Fv(A)1940 1309 y Fq(\030)1941 1341 y Fx(=)2045 1337 y(Hom)2248 1352 y Fr(A)2305 1337 y Fx(\()p Fv(:A;)17 b(:A)p Fx(\))2688 1280 y Fr(S)2653 1337 y Fq(\000)-60 b(!)28 b Fx(Hom)3001 1352 y Fr(B)3061 1337 y Fx(\()p Fv(:S)6 b(A;)17 b(:S)6 b(A)p Fx(\).)0 1482 y(10.)40 b Fv(T)14 b(B)342 1454 y Fq(\030)343 1486 y Fx(=)447 1482 y(Hom)651 1497 y Fr(B)711 1482 y Fx(\()p Fv(:S)6 b(A;)17 b(:B)5 b Fx(\),)25 b(since)e(Hom)1587 1497 y Fr(B)1647 1482 y Fx(\()p Fv(:S)6 b(A;)17 b(:B)5 b Fx(\))2100 1425 y Fr(T)2067 1482 y Fq(\000)-60 b(!)28 b Fx(Hom)2415 1497 y Fr(A)2472 1482 y Fx(\()p Fv(:T)14 b(S)6 b(A;)17 b(:T)d(B)5 b Fx(\))3033 1454 y Fq(\030)3034 1486 y Fx(=)3138 1482 y(Hom)3341 1497 y Fr(A)3398 1482 y Fx(\()p Fv(:A;)17 b(:T)d(B)5 b Fx(\))3823 1454 y Fq(\030)3823 1486 y Fx(=)0 1598 y Fv(T)14 b(B)5 b Fx(.)0 1714 y(11.)43 b(\()p Fv(B)5 b(;)17 b(A;)g(S)6 b(A;)17 b(T)d(B)5 b(;)17 b(f)5 b(;)17 b(g)t Fx(\))31 b(de\014nes)j(a)e(strict)h(Morita)g(con)m (text)h(b)m(y)f(Morita)g(I)s(I)s(I.)0 1831 y(12.)77 b(The)45 b(functor)f Fv(S)49 b Fx(is)c(isomorphic)g(to)e Fv(S)6 b(A)30 b Fq(\012)1864 1846 y Fr(A)1951 1831 y Fq(\000)p Fx(.)78 b(Infact)44 b(w)m(e)h(ha)m(v)m(e)g(Hom)3020 1846 y Fr(B)3081 1831 y Fx(\()p Fv(:S)6 b(A)30 b Fq(\012)3392 1846 y Fr(A)3479 1831 y Fv(M)5 b(;)17 b(:N)10 b Fx(\))3823 1803 y Fq(\030)3823 1835 y Fx(=)0 1947 y(Hom)203 1962 y Fr(A)260 1947 y Fx(\()p Fv(:M)5 b(;)17 b(:)g Fx(Hom)715 1962 y Fr(B)776 1947 y Fx(\()p Fv(:S)6 b(A;)17 b(:N)10 b Fx(\)\))1144 2035 y Fq(\030)1145 2067 y Fx(=)1249 2063 y(Hom)1452 2078 y Fr(A)1509 2063 y Fx(\()p Fv(:M)5 b(;)17 b(:)g Fx(Hom)1964 2078 y Fr(A)2021 2063 y Fx(\()p Fv(:A;)g(:T)d(N)c Fx(\)\))1144 2152 y Fq(\030)1145 2183 y Fx(=)1249 2179 y(Hom)1452 2194 y Fr(A)1509 2179 y Fx(\()p Fv(:M)5 b(;)17 b(:T)d(N)c Fx(\))1144 2268 y Fq(\030)1145 2300 y Fx(=)1249 2296 y(Hom)1452 2311 y Fr(B)1513 2296 y Fx(\()p Fv(:S)c(M)f(;)17 b(:N)10 b Fx(\))p Fv(:)0 2412 y Fx(The)34 b(represen)m(ting)g(ob)5 b(ject)1046 2427 y Fr(B)1107 2412 y Fv(S)h(M)1305 2384 y Fq(\030)1306 2416 y Fx(=)1411 2427 y Fr(B)1471 2412 y Fv(S)g(A)22 b Fq(\012)1709 2427 y Fr(A)1789 2412 y Fv(M)43 b Fx(dep)s(ends)34 b(functorially)g(on)e Fv(M)43 b Fx(b)m(y)34 b(5.5.)439 b Fo(\003)0 2586 y Ft(Prop)s(osition)44 b(7.12.)886 2601 y Fr(A)943 2586 y Fv(M)51 b Fs(is)39 b(\014nitely)h(gener)-5 b(ate)g(d)39 b(i\013)h(in)f(e)-5 b(ach)39 b(set)h(of)g(submo)-5 b(dules)39 b Fq(f)p Fv(A)3354 2601 y Fr(i)3382 2586 y Fq(j)p Fv(i)d Fq(2)i Fv(I)8 b Fq(g)39 b Fs(with)0 2702 y Fv(A)73 2717 y Fr(i)146 2702 y Fq(\022)44 b Fv(M)55 b Fs(and)614 2628 y Fl(P)719 2731 y Fr(i)p Fp(2)p Fr(I)847 2702 y Fv(A)920 2717 y Fr(i)992 2702 y Fx(=)45 b Fv(M)54 b Fs(ther)-5 b(e)44 b(is)g(a)f(\014nite)h (subset)g Fq(f)p Fv(A)2402 2717 y Fr(i)2430 2702 y Fq(j)p Fv(i)g Fq(2)h Fv(I)2689 2717 y Fk(0)2728 2702 y Fq(g)f Fs(\()p Fv(I)2905 2717 y Fk(0)2988 2702 y Fq(\022)h Fv(I)52 b Fs(\014nite\))43 b(such)h(that)0 2744 y Fl(P)105 2848 y Fr(i)p Fp(2)p Fr(I)207 2857 y Fg(0)262 2819 y Fv(A)335 2834 y Fr(i)391 2819 y Fx(=)28 b Fv(M)10 b Fs(.)0 2993 y(Pr)-5 b(o)g(of.)41 b Fx(Let)c Fv(M)47 b Fx(=)35 b Fv(Am)887 3008 y Fk(1)952 2993 y Fx(+)25 b Fv(:)17 b(:)g(:)25 b Fx(+)g Fv(Am)1452 3008 y Fr(n)1499 2993 y Fx(.)57 b(Eac)m(h)38 b Fv(m)1915 3008 y Fr(j)1989 2993 y Fx(is)g(con)m(tained)g(in)f(a)g (\014nite)h(sum)g(of)e(the)i Fv(A)3568 3008 y Fr(i)3596 2993 y Fx(,)h(hence)0 3109 y(all)i(of)h(the)f Fv(m)526 3124 y Fr(j)605 3109 y Fx(and)g(hence)i(the)f(mo)s(dule)g Fv(M)53 b Fx(itself.)71 b(Con)m(v)m(ersely)44 b(consider)f Fq(f)p Fv(Am)p Fq(j)p Fv(m)h Fq(2)f Fv(M)10 b Fq(g)p Fx(.)71 b(Then)0 3225 y Fv(M)38 b Fx(=)236 3151 y Fl(P)358 3225 y Fv(Am)p Fx(,)28 b(hence)h Fv(M)37 b Fx(is)28 b(a)e(sum)i(of)f (\014nitely)h(man)m(y)g(of)e(the)i Fv(Am)f Fx(and)g(th)m(us)h(is)g (\014nitely)g(generated.)99 b Fo(\003)0 3400 y Ft(Corollary)29 b(7.13.)36 b Fs(Under)27 b(an)f(e)-5 b(quivalenc)g(e)26 b(of)h(c)-5 b(ate)g(gories)26 b Fv(T)41 b Fx(:)28 b Fv(A)p Fs(-)p Fx(Mo)s(d)g Fq(\000)-57 b(!)27 b Fv(B)5 b Fs(-)p Fx(Mo)s(d)27 b Fs(\014nitely)g(gener)-5 b(ate)g(d)0 3516 y(mo)g(dules)34 b(ar)-5 b(e)35 b(mapp)-5 b(e)g(d)33 b(into)i (\014nitely)g(gener)-5 b(ate)g(d)34 b(mo)-5 b(dules.)0 3690 y(Pr)g(o)g(of.)41 b Fx(The)35 b(lattice)f(of)g(submo)s(dules)h Fq(V)8 b Fx(\()p Fv(M)i Fx(\))35 b(is)f(isomorphic)h(to)f(the)g (lattice)g(of)g(submo)s(dules)h Fq(V)8 b Fx(\()p Fv(T)14 b(M)c Fx(\).)3823 3807 y Fo(\003)0 3981 y Ft(Problem)35 b(7.1.)41 b Fx(Let)30 b Fv(A)p Fx(-Mo)s(d)h(b)s(e)f(equiv)-5 b(alen)m(t)33 b(to)d Fv(B)5 b Fx(-Mo)s(d.)42 b(Sho)m(w)32 b(that)e(Mo)s(d-)p Fv(A)g Fx(and)h(Mo)s(d-)p Fv(B)k Fx(are)c(also)0 4097 y(equiv)-5 b(alen)m(t.)0 4272 y Ft(Problem)37 b(7.2.)42 b Fx(Sho)m(w)33 b(that)e(an)h(equiv)-5 b(alence)35 b(of)c(arbitrary)h (categories)h(preserv)m(es)i(monomorphisms.)0 4446 y Ft(Problem)e(7.3.)39 b Fx(Sho)m(w)30 b(that)e(an)h(equiv)-5 b(alence)31 b(of)d(mo)s(dule)h(categories)h(preserv)m(es)h(pro)5 b(jectiv)m(e)31 b(mo)s(dules,)0 4562 y(but)i(not)f(free)h(mo)s(dules.)p eop end %%Page: 66 66 TeXDict begin 66 65 bop 0 -170 a Fu(66)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)830 29 y Fx(8.)48 b Fw(Simple)38 b(and)g(Semisimple)f(rings)g(and)h(Modules)0 204 y Fx(8.1.)49 b Ft(Simple)38 b(and)h(Semisimple)g(rings.)0 379 y(De\014nition)f(8.1.) k Fx(An)33 b(ideal)1132 394 y Fr(R)1190 379 y Fv(I)i Fq(\022)1373 394 y Fr(R)1431 379 y Fv(R)f Fx(is)f(called)g Fs(nilp)-5 b(otent)p Fx(,)32 b(if)h(there)g(is)g Fv(n)28 b Fq(\025)g Fx(1)k(suc)m(h)i(that)f Fv(I)3533 343 y Fr(n)3607 379 y Fx(=)28 b(0.)0 495 y(A)22 b(mo)s(dule)429 510 y Fr(R)486 495 y Fv(M)33 b Fx(is)23 b(called)g Fs(A)n(rtinian)e Fx(\(Emil)i(Artin,)h(1898-1962\),)e(if)g(eac)m(h)h(non)f(empt)m(y)i (set)e(of)g(submo)s(dules)0 612 y(of)32 b Fv(M)43 b Fx(con)m(tains)34 b(a)e(minimal)i(elemen)m(t.)0 728 y(A)39 b(mo)s(dule)462 743 y Fr(R)519 728 y Fv(M)50 b Fx(is)39 b(called)g Fs(No)-5 b(etherian)38 b Fx(\(Emm)m(y)i(No)s(ether,)h(1882-1935\),)d(if)g(eac)m (h)h(non)g(empt)m(y)h(set)f(of)0 844 y(submo)s(dules)34 b(of)f Fv(M)43 b Fx(con)m(tains)33 b(a)g(maximal)g(elemen)m(t.)0 960 y(A)i(ring)f Fv(R)i Fx(is)f(called)g Fs(simple)p Fx(,)f(if)1221 975 y Fr(R)1278 960 y Fv(R)i Fx(as)f(a)f(mo)s(dule)h(is) g(Artinian)g(and)g(if)f Fv(R)i Fx(do)s(es)f(not)f(ha)m(v)m(e)i(non)f (trivial)0 1076 y(\()p Fq(6)p Fx(=)27 b(0)p Fv(;)17 b(R)q Fx(\))32 b(t)m(w)m(o)i(sided)g(ideals.)0 1193 y(A)39 b(ring)g Fv(R)h Fx(is)g(called)g Fs(semisimple)p Fx(,)f(if)1448 1208 y Fr(R)1506 1193 y Fv(R)h Fx(is)f(Artinian)h(and)f(if)g Fv(R)h Fx(do)s(es)g(not)f(ha)m(v)m(e)h(non)f(trivial)h(\()p Fq(6)p Fx(=)e(0\))0 1309 y(nilp)s(oten)m(t)33 b(left)g(ideals.)0 1484 y Ft(Lemma)39 b(8.2.)j Fs(Each)34 b(simple)g(ring)h(is)f (semisimple.)0 1659 y(Pr)-5 b(o)g(of.)41 b Fv(C)j Fx(:=)553 1585 y Fl(P)658 1659 y Fx(\()p Fv(I)8 b Fq(j)775 1674 y Fr(R)832 1659 y Fv(I)45 b Fq(\022)1035 1674 y Fr(R)1093 1659 y Fv(R)39 b Fx(nilp)s(oten)m(t\))g(is)g(a)f(t)m(w)m(o)g(sided)i (ideal.)61 b(In)38 b(fact)h(tak)m(e)g Fv(a)e Fq(2)h Fv(I)46 b Fx(and)38 b Fv(r)i Fq(2)e Fv(R)q Fx(.)0 1776 y(Then)707 1893 y(\()p Fv(r)789 1908 y Fk(1)828 1893 y Fv(ar)s Fx(\)\()p Fv(r)1046 1908 y Fk(2)1085 1893 y Fv(ar)s Fx(\))17 b Fv(:)g(:)g(:)f Fx(\()p Fv(r)1451 1908 y Fr(n)1498 1893 y Fv(ar)s Fx(\))27 b(=)h(\()p Fv(r)1847 1908 y Fk(1)1886 1893 y Fv(a)p Fx(\)\()p Fv(r)s(r)2104 1908 y Fk(2)2143 1893 y Fv(a)p Fx(\))17 b Fv(:)g(:)g(:)f Fx(\()p Fv(r)s(r)2509 1908 y Fr(n)2556 1893 y Fv(a)p Fx(\))p Fv(r)30 b Fq(2)e Fv(I)2864 1852 y Fr(n)2911 1893 y Fv(R)h Fx(=)e(0)p Fv(:)0 2032 y Fx(Hence)32 b(w)m(e)f(ha)m(v)m(e)h(\()p Fv(R)q(ar)s Fx(\))901 1995 y Fr(n)976 2032 y Fx(=)27 b(0)h(=)-17 b Fq(\))27 b Fv(R)q(ar)k Fq(\022)d Fv(C)7 b Fx(,)31 b(so)g Fv(ar)f Fq(2)e Fv(C)38 b Fx(and)30 b Fv(C)38 b Fx(is)31 b(a)f(t)m(w)m(o)h(sided)h(ideal.)43 b(Th)m(us)32 b Fv(C)j Fx(=)27 b(0)0 2148 y(or)39 b Fv(C)47 b Fx(=)40 b Fv(R)q Fx(.)66 b(If)39 b Fv(C)47 b Fx(=)40 b(0)g(then)g(there)h(are)f(no)f (non)h(trivial)h(nilp)s(oten)m(t)f(ideals.)66 b(If)40 b Fv(C)47 b Fx(=)40 b Fv(R)h Fx(then)f(there)0 2264 y(are)d(ideals)h (and)f(elemen)m(ts)j Fv(a)1092 2279 y Fr(i)1156 2264 y Fq(2)35 b Fv(I)1300 2279 y Fr(i)1366 2264 y Fx(suc)m(h)j(that)f(1)e (=)h Fv(a)2053 2279 y Fk(1)2117 2264 y Fx(+)26 b Fv(:)17 b(:)g(:)24 b Fx(+)h Fv(a)2510 2279 y Fr(n)2558 2264 y Fx(.)57 b(The)38 b(ideal)f Fv(I)3127 2279 y Fk(1)3192 2264 y Fx(+)25 b Fv(I)3336 2279 y Fk(2)3413 2264 y Fx(is)37 b(nilp)s(oten)m(t)0 2380 y(since)c(\()p Fv(a)327 2395 y Fk(1)385 2380 y Fx(+)20 b Fv(b)522 2395 y Fk(1)561 2380 y Fx(\)\()p Fv(a)688 2395 y Fk(2)747 2380 y Fx(+)f Fv(b)883 2395 y Fk(2)923 2380 y Fx(\))e Fv(:)g(:)g(:)f Fx(\()p Fv(a)1198 2395 y Fk(2)p Fr(n)1300 2380 y Fx(+)j Fv(b)1436 2395 y Fk(2)p Fr(n)1518 2380 y Fx(\))31 b(consists)i(of)e (monomials)h(either)g(in)f Fv(I)2980 2344 y Fr(n)2972 2405 y Fk(1)3046 2380 y Fq(\001)19 b Fv(R)33 b Fx(or)d(in)i Fv(I)3481 2344 y Fr(n)3473 2405 y Fk(2)3547 2380 y Fq(\001)19 b Fv(R)q Fx(.)43 b(But)0 2497 y Fv(I)51 2460 y Fr(n)43 2521 y Fk(1)125 2497 y Fx(=)28 b(0)f(=)h Fv(I)460 2460 y Fr(n)452 2521 y Fk(2)535 2497 y Fx(=)-17 b Fq(\))27 b Fx(\()p Fv(I)802 2512 y Fk(1)864 2497 y Fx(+)22 b Fv(I)1005 2512 y Fk(2)1044 2497 y Fx(\))1082 2460 y Fk(2)p Fr(n)1192 2497 y Fx(=)28 b(0.)43 b(Hence)34 b(1)e(is)h(nilp)s(oten)m(t.)44 b(Con)m(tradiction.)870 b Fo(\003)0 2672 y Ft(De\014nition)44 b(8.3.)g Fx(A)37 b(mo)s(dule)1206 2687 y Fr(R)1264 2672 y Fv(M)48 b Fx(is)38 b(called)g Fs(simple)f Fx(i\013)g Fv(M)47 b Fq(6)p Fx(=)36 b(0)h(and)g Fv(M)48 b Fx(has)38 b(only)g(the)g(mo)s(dules)g(0)0 2788 y(and)33 b Fv(M)43 b Fx(as)33 b(submo)s(dules.)45 b(An)33 b(ideal)1407 2803 y Fr(R)1465 2788 y Fv(I)40 b Fx(is)33 b(called)h Fs(simple)d Fx(or)h Fs(minimal)p Fx(,)g(if)g(it)h(is)g(simple)h(as)f(a)f(mo)s (dule.)0 2963 y Ft(Lemma)39 b(8.4.)j Fs(L)-5 b(et)35 b Fv(R)h Fs(b)-5 b(e)35 b(semisimple.)43 b(Then)34 b(e)-5 b(ach)34 b(left)h(ide)-5 b(al)34 b(of)g Fv(R)i Fs(is)f(a)g(dir)-5 b(e)g(ct)34 b(summand)g(of)h Fv(R)q Fs(.)0 3139 y(Pr)-5 b(o)g(of.)41 b Fx(Let)h Fv(I)49 b Fx(b)s(e)42 b(an)f(ideal)h(in)g Fv(R)q Fx(,)i(that)d(is)h(not)g(a)f(direct)h(summand,)j(and)d(let)g Fv(I)49 b Fx(b)s(e)42 b(minimal)g(with)0 3255 y(resp)s(ect)34 b(to)e(this)h(prop)s(ert)m(y)-8 b(.)44 b(Suc)m(h)34 b(an)f(ideal)g (exists,)h(since)g Fv(R)g Fx(Artinian.)0 3371 y(Case)d(1:)42 b(Let)30 b Fv(I)36 b Fq(\022)28 b Fv(R)j Fx(b)s(e)f(an)g(ideal)h(that)e (is)i(not)f(minimal)h(\(simple\),)h(i.e.)43 b(there)31 b(is)f(an)g(ideal)h Fv(J)36 b Fq(\022)28 b Fv(I)38 b Fx(with)0 3487 y(0)f Fq(6)p Fx(=)f Fv(J)46 b Fq(6)p Fx(=)37 b Fv(I)8 b Fx(.)59 b(Then)39 b Fv(J)47 b Fx(is)39 b(a)e(direct)i (summand)g(of)f Fv(R)q Fx(,)h(i.e.)60 b(there)39 b(is)f(a)g (homomorphism)h Fv(f)47 b Fx(:)37 b Fv(R)h Fq(\000)-60 b(!)37 b Fv(J)0 3631 y Fx(with)d(\()p Fv(J)39 b Fq(\000)-60 b(!)29 b Fv(I)37 b Fq(\000)-59 b(!)28 b Fv(R)869 3571 y Fr(f)831 3631 y Fq(\000)-60 b(!)29 b Fv(J)9 b Fx(\))30 b(=)f(id)1294 3646 y Fr(J)1343 3631 y Fx(.)46 b(This)35 b(implies)g Fv(I)i Fx(=)29 b Fv(J)j Fq(\010)24 b Fv(K)40 b Fx(for)33 b Fv(K)k Fx(:=)29 b(Ke\()p Fv(I)37 b Fq(\000)-59 b(!)29 b Fv(R)3395 3571 y Fr(f)3357 3631 y Fq(\000)-60 b(!)29 b Fv(J)9 b Fx(\).)47 b(Since)0 3764 y Fv(K)35 b Fq(6)p Fx(=)27 b Fv(I)8 b Fx(,)31 b(there)g(is)f(also)h(a)e Fv(g)i Fx(:)d Fv(R)h Fq(\000)-60 b(!)28 b Fv(K)37 b Fx(with)31 b(\()p Fv(K)j Fq(\000)-59 b(!)27 b Fv(I)36 b Fq(\000)-60 b(!)27 b Fv(R)2332 3703 y Fr(g)2291 3764 y Fq(\000)-60 b(!)28 b Fv(K)7 b Fx(\))28 b(=)f(id)2776 3779 y Fr(K)2845 3764 y Fx(.)42 b(The)31 b(map)g Fv(f)d Fx(+)17 b Fv(g)j Fq(\000)e Fv(g)t(f)37 b Fx(:)28 b Fv(I)0 3880 y Fq(\000)-60 b(!)28 b Fv(R)g Fq(\000)-59 b(!)27 b Fv(I)39 b Fx(satis\014es)33 b(\()p Fv(f)e Fx(+)19 b Fv(g)k Fq(\000)d Fv(g)t(f)11 b Fx(\)\()p Fv(j)6 b Fx(\))26 b(=)i Fv(f)11 b Fx(\()p Fv(j)6 b Fx(\))19 b(+)g Fv(g)t Fx(\()p Fv(j)6 b Fx(\))19 b Fq(\000)h Fv(g)t(f)11 b Fx(\()p Fv(j)6 b Fx(\))26 b(=)i Fv(j)e Fx(+)19 b Fv(g)t Fx(\()p Fv(j)6 b Fx(\))19 b Fq(\000)h Fv(g)t Fx(\()p Fv(j)6 b Fx(\))26 b(=)i Fv(j)37 b Fx(for)31 b(all)g Fv(j)j Fq(2)28 b Fv(J)0 3996 y Fx(and)g(\()p Fv(f)22 b Fx(+)12 b Fv(g)k Fq(\000)c Fv(g)t(f)f Fx(\)\()p Fv(k)s Fx(\))27 b(=)g Fv(f)11 b Fx(\()p Fv(k)s Fx(\))h(+)g Fv(g)t Fx(\()p Fv(k)s Fx(\))g Fq(\000)g Fv(g)t(f)f Fx(\()p Fv(k)s Fx(\))26 b(=)h(0)12 b(+)g Fv(k)j Fq(\000)d Fx(0)28 b(=)g Fv(k)i Fx(for)d(all)h Fv(k)j Fq(2)d Fv(K)7 b Fx(,)28 b(hence)h(\()p Fv(f)23 b Fx(+)12 b Fv(g)k Fq(\000)c Fv(g)t(f)37 b Fx(:)28 b Fv(I)0 4112 y Fq(\000)-60 b(!)28 b Fv(R)g Fq(\000)-59 b(!)27 b Fv(I)8 b Fx(\))28 b(=)f(id)694 4127 y Fr(I)733 4112 y Fx(.)44 b(Th)m(us)34 b Fv(I)40 b Fx(is)33 b(a)g(direct)g(summand)h(of)e Fv(R)q Fx(.)44 b(Con)m(tradiction.)0 4229 y(Case)35 b(2:)47 b(Let)35 b Fv(I)42 b Fx(b)s(e)35 b(a)f(minimal)h(or)f(simple)i(ideal.)49 b(Since)36 b Fv(I)42 b Fx(is)35 b(not)f(nilp)s(oten)m(t)h(and)g(0)30 b Fq(6)p Fx(=)h Fv(I)3387 4193 y Fk(2)3457 4229 y Fq(\022)g Fv(I)42 b Fx(holds,)0 4345 y(w)m(e)28 b(get)f Fv(I)346 4309 y Fk(2)413 4345 y Fx(=)h Fv(I)8 b Fx(.)41 b(In)27 b(particular)h(there)g(exists)g(an)f Fv(a)h Fq(2)g Fv(I)35 b Fx(with)28 b Fv(I)8 b(a)28 b Fx(=)f Fv(I)8 b Fx(,)28 b(since)h Fv(I)8 b(a)27 b Fx(is)g(also)g(an)g(ideal.)42 b(Th)m(us)0 4461 y Fq(\001)p Fv(a)28 b Fx(:)f Fv(I)36 b Fq(\000)-60 b(!)28 b Fv(I)39 b Fx(is)32 b(an)f(epimorphism)i(and)e (ev)m(en)i(an)f(isomorphism,)h(for)e(Ke\()p Fq(\001)p Fv(a)p Fx(\))g(m)m(ust)h(b)s(e)g(zero)g(as)f(an)g(ideal)0 4577 y(\(see)c(Lemma)g(of)e(Sc)m(h)m(ur)i(8.5.\))41 b(So)26 b(there)h(is)f(an)g Fv(e)i Fq(2)g Fv(I)8 b Fx(,)27 b Fv(e)h Fq(6)p Fx(=)g(0)d(with)i Fv(ea)h Fx(=)f Fv(a:)h Fx(=)-17 b Fq(\))28 b Fx(\()p Fv(e)3064 4541 y Fk(2)3112 4577 y Fq(\000)9 b Fv(e)p Fx(\))p Fv(a)28 b Fx(=)g Fv(eea)9 b Fq(\000)g Fv(ea)28 b Fx(=)0 4694 y Fv(a)21 b Fq(\000)f Fv(a)28 b Fx(=)g(0)f(=)-17 b Fq(\))28 b Fv(e)660 4657 y Fk(2)720 4694 y Fq(\000)20 b Fv(e)28 b Fx(=)g(0)f Fq(2)h Fv(I)36 b Fx(=)-17 b Fq(\))28 b Fv(e)1475 4657 y Fk(2)1542 4694 y Fx(=)f Fv(e)h Fq(2)g Fv(I)8 b Fx(.)43 b(F)-8 b(rom)32 b Fv(I)j Fx(=)28 b Fv(R)q(e)k Fx(w)m(e)g(get)g Fv(R)d Fx(=)e Fv(R)q(e)21 b Fq(\010)g Fv(R)q Fx(\(1)f Fq(\000)g Fv(e)p Fx(\),)33 b(since)0 4810 y Fv(R)c Fx(=)e Fv(R)q(e)20 b Fx(+)g Fv(R)q Fx(\(1)f Fq(\000)h Fv(e)p Fx(\))31 b(and)h Fv(r)s(e)27 b Fx(=)h Fv(s)p Fx(\(1)19 b Fq(\000)h Fv(e)p Fx(\))28 b Fq(2)g Fv(R)q(e)20 b Fq(\\)g Fv(R)q Fx(\(1)f Fq(\000)h Fv(e)p Fx(\))28 b(=)-17 b Fq(\))28 b Fv(r)s(e)f Fx(=)h Fv(r)s(e)2817 4774 y Fk(2)2884 4810 y Fx(=)f Fv(s)p Fx(\(1)20 b Fq(\000)g Fv(e)p Fx(\))p Fv(e)28 b Fx(=)f(0)p Fv(:)k Fx(Th)m(us)i Fv(I)0 4926 y Fx(is)g(a)f(direct)i(summand)g(of)e Fv(R)q Fx(.)43 b(Con)m(tradiction.)2052 b Fo(\003)0 5101 y Ft(Lemma)39 b(8.5.)j Fx(\(Sc)m(h)m(ur\))36 b Fs(L)-5 b(et)1130 5116 y Fr(R)1188 5101 y Fv(M)10 b Fs(,)1358 5116 y Fr(R)1415 5101 y Fv(N)46 b Fs(b)-5 b(e)34 b(simple)g(mo)-5 b(dules.)44 b(Then)34 b(the)h(fol)5 b(lowing)34 b(hold:)148 5239 y Fx(\(1\))42 b Fs(If)34 b Fv(M)k Fq(6)549 5211 y(\030)550 5243 y Fx(=)654 5239 y Fv(N)10 b Fs(,)36 b(then)e Fx(Hom)1228 5254 y Fr(R)1285 5239 y Fx(\()p Fv(:M)5 b(;)17 b(:N)10 b Fx(\))29 b(=)e(0)p Fv(:)148 5355 y Fx(\(2\))42 b(Hom)518 5370 y Fr(R)575 5355 y Fx(\()p Fv(:M)5 b(;)17 b(:M)10 b Fx(\))36 b Fs(is)f(a)g(skew-\014eld)e(\(=)h(division)g (algebr)-5 b(a)34 b(=)h(non)f(c)-5 b(ommutative)34 b(\014eld\).)0 5530 y(Pr)-5 b(o)g(of.)41 b Fx(Let)h Fv(f)54 b Fx(:)43 b Fv(M)54 b Fq(\000)-60 b(!)43 b Fv(N)52 b Fx(b)s(e)42 b(a)f(homomorphism)i(with)f Fv(f)54 b Fq(6)p Fx(=)43 b(0.)70 b(Then)43 b(Im\()p Fv(f)11 b Fx(\))43 b(=)g Fv(N)10 b Fx(,)44 b(since)f Fv(N)52 b Fx(is)0 5646 y(simple)44 b(and)f(Ke\()p Fv(f)11 b Fx(\))44 b(=)g(0,)h(since)f Fv(M)53 b Fx(is)43 b(simple,)k(hence)d Fv(f)53 b Fx(is)43 b(an)g(isomorphism.)75 b(This)43 b(implies)h(\(1\).)p eop end %%Page: 67 67 TeXDict begin 67 66 bop 1239 -170 a Fu(Simple)26 b(and)f(Semisimple)h (rings)h(and)e(Mo)r(dules)1163 b(67)0 29 y Fx(F)-8 b(urthermore)39 b(w)m(e)g(ha)m(v)m(e)h(\(2\),)f(since)h(eac)m(h)f(endomorphism)h Fv(f)48 b Fx(:)37 b Fv(M)48 b Fq(\000)-59 b(!)37 b Fv(M)49 b Fx(with)39 b Fv(f)48 b Fq(6)p Fx(=)37 b(0)h(is)h(in)m(v)m(ertible)0 146 y(under)e(the)g(m)m(ultiplication)g(of)f(Hom)1399 161 y Fr(R)1457 146 y Fx(\()p Fv(:M)5 b(;)17 b(:M)10 b Fx(\).)55 b(Observ)m(e)38 b(that)e(a)f Fs(skew-\014eld)g Fx(is)i(a)e(ring,)j(whose)f(non)0 262 y(zero)c(elemen)m(ts)i(form)d(a)g (group)h(under)g(the)g(m)m(ultiplication.)1567 b Fo(\003)0 483 y Ft(Remark)31 b(8.6.)38 b Fx(Let)799 498 y Fr(R)856 483 y Fv(M)g Fx(b)s(e)27 b(simple.)43 b(Then)28 b(End)1879 498 y Fr(R)1936 483 y Fx(\()p Fv(:M)10 b Fx(\))29 b(=)e Fv(D)j Fx(is)d(a)g(sk)m(ew-\014eld.)43 b(Hence)29 b(the)e Fv(R)q Fx(-mo)s(dule)0 599 y(structure)34 b(of)e Fv(M)43 b Fx(can)33 b(b)s(e)g(c)m(haracterized)h(b)m(y)g Fv(R)29 b Fq(\000)-60 b(!)27 b Fx(End)2129 614 y Fr(D)2193 599 y Fx(\()p Fv(M)5 b(:)p Fx(\))28 b(=)g Fv(M)2621 614 y Fr(n)2668 599 y Fx(\()p Fv(D)s Fx(\))p Fv(:)0 819 y Ft(Theorem)38 b(8.7.)k Fx(\(Artin-W)-8 b(edderburn\))36 b Fs(The)e(fol)5 b(lowing)34 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)148 996 y Fx(\(1\))42 b Fv(R)35 b Fs(is)g(simple.)148 1113 y Fx(\(2\))42 b Fv(R)35 b Fs(p)-5 b(ossesses)34 b(a)h(simple)e(ide)-5 b(al)35 b(that)g(is)f(an)h Fv(R)q Fs(-pr)-5 b(o)g(gener)g(ator.)148 1229 y Fx(\(3\))42 b Fv(R)421 1201 y Fq(\030)422 1233 y Fx(=)530 1229 y Fv(M)624 1244 y Fr(n)671 1229 y Fx(\()p Fv(D)s Fx(\))37 b Fs(is)g(a)f(ful)5 b(l)37 b(matrix)g(ring)f(over)h(a)g (skew-\014eld)e Fv(D)s Fs(.)51 b(\()p Fv(n)37 b Fs(is)f(unique,)i Fv(D)h Fs(is)e(unique)g(up)315 1345 y(to)e(isomorphism.\))148 1461 y Fx(\(4\))42 b Fv(R)28 b Fx(=)g Fv(I)564 1476 y Fk(1)625 1461 y Fq(\010)23 b Fv(:)17 b(:)g(:)22 b Fq(\010)g Fv(I)1004 1476 y Fr(n)1086 1461 y Fs(with)35 b(isomorphic)e(simple)h (left)h(ide)-5 b(als)34 b Fv(I)2580 1476 y Fk(1)2619 1461 y Fv(;)17 b(:)g(:)g(:)f(;)h(I)2881 1476 y Fr(n)2928 1461 y Fs(.)0 1682 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))32 b(=)-17 b Fq(\))33 b Fx(\(2\):)43 b(Since)34 b Fv(R)g Fx(is)f(Artinian)g(there)g(is)g(a)g(simple)h(ideal)f(0)28 b Fq(6)p Fx(=)g Fv(I)35 b Fq(\022)29 b Fv(R)q Fx(.)43 b(Let)33 b Fv(J)k Fx(:=)3569 1607 y Fl(P)3674 1682 y Fq(f)p Fv(I)3775 1646 y Fp(0)3798 1682 y Fq(j)p Fv(I)3877 1646 y Fp(0)0 1798 y Fx(ideal)g(in)f Fv(R)h Fx(and)g Fv(I)710 1762 y Fp(0)767 1770 y Fq(\030)768 1802 y Fx(=)878 1798 y Fv(I)8 b Fq(g)p Fx(.)54 b(Then)38 b Fv(J)45 b Fx(is)37 b(a)f(t)m(w)m(o)h(sided)g(ideal,)h(since)g Fv(I)2605 1762 y Fp(0)2652 1798 y Fq(\001)25 b Fv(r)36 b Fq(6)p Fx(=)e(0)g(=)-17 b Fq(\))34 b(\001)p Fv(r)i Fx(:)e Fv(I)3391 1762 y Fp(0)3448 1798 y Fq(\000)-59 b(!)33 b Fv(R)k Fx(with)0 1914 y(Ke\()p Fq(\001)p Fv(r)s Fx(\))30 b(=)g(0,)35 b(hence)g Fq(\001)p Fv(r)i Fx(is)e(injectiv)m(e)h(and)e(the)h(image)f Fv(I)2082 1878 y Fp(0)2128 1914 y Fq(\001)23 b Fv(r)37 b Fx(is)e(isomorphic)g(to)f Fv(I)3027 1878 y Fp(0)3084 1914 y Fx(resp.)50 b Fv(I)8 b Fx(,)34 b(hence)i(is)f(in)0 2031 y Fv(J)9 b Fx(.)42 b(Since)30 b Fv(R)g Fx(is)g(simple)g(w)m(e)g (ha)m(v)m(e)g Fv(R)f Fx(=)e Fv(J)37 b Fx(=)1643 1956 y Fl(P)1765 2031 y Fv(I)1808 2046 y Fr(i)1836 2031 y Fx(.)43 b(Since)30 b(1)d Fq(2)h Fv(I)2370 2046 y Fk(1)2424 2031 y Fx(+)14 b Fv(:)j(:)g(:)d Fx(+)g Fv(I)2776 2046 y Fr(n)2824 2031 y Fx(,)29 b(there)h(is)f(an)g(epimorphism)0 2147 y Fv(I)43 2162 y Fk(1)111 2147 y Fq(\010)g Fv(:)17 b(:)g(:)28 b Fq(\010)h Fv(I)509 2162 y Fr(n)599 2147 y Fq(\000)-60 b(!)43 b Fv(R)g Fx(\(exterior)g(direct)f(sum\),)j(that)d (splits)h(since)g Fv(R)g Fx(is)f(pro)5 b(jectiv)m(e.)73 b(Hence)43 b Fv(R)g Fx(is)f(a)0 2263 y(direct)32 b(summand)h(of)e Fv(I)872 2278 y Fk(1)931 2263 y Fq(\010)20 b Fv(:)d(:)g(:)j Fq(\010)g Fv(I)1303 2278 y Fr(n)1381 2263 y Fx(up)32 b(to)f(isomorphism,)i(and)f(th)m(us)g Fv(I)39 b Fx(is)32 b(a)f(generator.)44 b(F)-8 b(urthermore)0 2379 y Fv(I)49 b Fx(is)42 b(a)f(direct)h(summand)g(of)f Fv(R)h Fx(b)m(y)g(8.4,)h (hence)g(it)e(is)h(\014nitely)h(generated)f(pro)5 b(jectiv)m(e,)45 b(th)m(us)d Fv(I)49 b Fx(is)42 b(an)0 2495 y Fv(R)q Fx(-progenerator.)0 2612 y(\(2\))j(=)-17 b Fq(\))44 b Fx(\(3\):)68 b(By)46 b(the)f(Lemma)h(of)e(Sc)m(h)m(ur)j(End)1881 2627 y Fr(R)1939 2612 y Fx(\()p Fv(:I)8 b Fx(\))49 b(=:)f Fv(D)g Fx(is)e(a)e(sk)m (ew-\014eld.)3146 2627 y Fr(R)3204 2612 y Fv(I)3247 2627 y Fr(D)3356 2612 y Fx(generates)i(an)0 2728 y(equiv)-5 b(alence)35 b(of)d(categories.)44 b(Hence)34 b Fv(R)1507 2700 y Fq(\030)1508 2732 y Fx(=)1612 2728 y(End)1787 2743 y Fr(D)1851 2728 y Fx(\()p Fv(I)8 b(:)p Fx(\))2032 2700 y Fq(\030)2033 2732 y Fx(=)2137 2728 y Fv(M)2231 2743 y Fr(n)2279 2728 y Fx(\()p Fv(D)s Fx(\))p Fv(:)0 2844 y Fx(\(3\))35 b(=)-17 b Fq(\))35 b Fx(\(4\):)49 b Fv(R)662 2817 y Fq(\030)663 2848 y Fx(=)772 2844 y Fv(M)866 2859 y Fr(n)913 2844 y Fx(\()p Fv(D)s Fx(\))32 b(=)-17 b Fq(\))32 b Fv(R)1404 2817 y Fq(\030)1405 2848 y Fx(=)1514 2844 y(End)1689 2859 y Fr(D)1753 2844 y Fx(\()p Fv(V)5 b(:)p Fx(\))35 b(with)h(an)f Fv(n)p Fx(-dimensional)i Fv(D)s Fx(-v)m(ector)e(space)i Fv(V)21 b Fx(.)52 b Fv(V)3836 2859 y Fr(D)0 2960 y Fx(is)42 b(a)f(progenerator.)71 b(Hence)43 b(w)m(e)g(ha)m(v)m(e)g Fq(V)8 b Fx(\()1630 2975 y Fr(R)1687 2960 y Fv(R)q Fx(\))1843 2933 y Fq(\030)1844 2965 y Fx(=)1964 2960 y Fq(V)g Fx(\()2071 2975 y Fr(D)2135 2960 y Fv(V)2214 2924 y Fp(\003)2253 2960 y Fx(\).)71 b(Since)42 b Fv(V)2731 2924 y Fp(\003)2813 2933 y Fq(\030)2814 2965 y Fx(=)2934 2960 y Fv(D)31 b Fq(\010)e Fv(:)17 b(:)g(:)28 b Fq(\010)g Fv(D)s Fx(,)44 b(w)m(e)f(ha)m(v)m(e)0 3092 y Fr(R)58 3077 y Fv(R)160 3049 y Fq(\030)161 3081 y Fx(=)266 3077 y Fv(I)309 3092 y Fk(1)370 3077 y Fq(\010)23 b Fv(:)17 b(:)g(:)k Fq(\010)i Fv(I)749 3092 y Fr(n)829 3077 y Fx(with)33 b Fv(I)1094 3092 y Fk(1)1161 3049 y Fq(\030)1162 3081 y Fx(=)1266 3077 y Fv(:)17 b(:)g(:)1408 3049 y Fq(\030)1409 3081 y Fx(=)1513 3077 y Fv(I)1556 3092 y Fr(n)1631 3049 y Fq(\030)1632 3081 y Fx(=)1736 3092 y Fr(R)1794 3077 y Fv(V)44 b Fq(\012)1972 3092 y Fr(D)2058 3077 y Fv(D)2170 3049 y Fq(\030)2170 3081 y Fx(=)2275 3092 y Fr(R)2333 3077 y Fv(V)21 b Fx(.)0 3193 y(\(4\))32 b(=)-17 b Fq(\))33 b Fx(\(2\):)43 b Fv(I)587 3208 y Fk(1)659 3193 y Fx(is)33 b(ob)m(viously)h(an)f Fv(R)q Fx(-progenerator.)0 3309 y(\(2\))h(=)-17 b Fq(\))35 b Fx(\(1\):)47 b Fv(R)q Fx(-Mo)s(d)886 3281 y Fq(\030)887 3313 y Fx(=)995 3309 y Fv(D)s Fx(-)o(Mo)s(d)35 b(with)g Fv(D)1680 3281 y Fq(\030)1681 3313 y Fx(=)1789 3309 y(End)1963 3324 y Fr(R)2021 3309 y Fx(\()p Fv(I)8 b Fx(\).)50 b(Hence)36 b Fq(V)8 b Fx(\()2624 3324 y Fr(R)2682 3309 y Fv(R)q Fx(\))2826 3281 y Fq(\030)2827 3313 y Fx(=)2935 3309 y Fq(V)g Fx(\()3042 3324 y Fr(D)3122 3309 y Fx(Hom)3326 3324 y Fr(D)3390 3309 y Fx(\()p Fv(I)g(:;)3550 3324 y Fr(D)3613 3309 y Fv(D)s(:)p Fx(\)\))34 b(is)0 3425 y(Artinian,)f(and)g (w)m(e)g(ha)m(v)m(e)h Fq(V)8 b Fx(\()1085 3440 y Fr(R)1143 3425 y Fv(R)1217 3440 y Fr(R)1275 3425 y Fx(\))1341 3398 y Fq(\030)1341 3429 y Fx(=)1446 3425 y Fq(V)g Fx(\()1553 3440 y Fr(D)1617 3425 y Fv(D)1698 3440 y Fr(D)1762 3425 y Fx(\))28 b(=)f Fq(f)p Fx(0)p Fv(;)17 b(D)s Fq(g)p Fx(.)42 b(Th)m(us)35 b Fv(R)e Fx(is)g(simple.)796 b Fo(\003)0 3646 y Ft(Corollary)51 b(8.8.)c Fs(L)-5 b(et)45 b Fv(R)g Fs(b)-5 b(e)44 b(a)g(simple)g(ring)f(and)h(let)2126 3661 y Fr(R)2183 3646 y Fv(M)56 b Fq(6)p Fx(=)45 b(0)f Fs(b)-5 b(e)44 b(\014nitely)g(gener)-5 b(ate)g(d.)73 b(Then)43 b(the)0 3762 y(fol)5 b(lowing)33 b(hold)148 3939 y Fx(\(1\))315 3954 y Fr(R)372 3939 y Fv(M)46 b Fs(is)35 b(an)f Fv(R)q Fs(-pr)-5 b(o)g(gener)g(ator.)148 4055 y Fx(\(2\))42 b Fv(S)33 b Fx(:=)28 b(End)713 4070 y Fr(R)771 4055 y Fx(\()p Fv(:M)10 b Fx(\))36 b Fs(is)e(a)h(simple)f(ring.)148 4172 y Fx(\(3\))42 b(Cen)m(t)q(\()p Fv(R)q Fx(\))696 4144 y Fq(\030)697 4176 y Fx(=)802 4172 y(Cen)m(t)q(\(End)1217 4187 y Fr(R)1275 4172 y Fx(\()p Fv(:M)10 b Fx(\)\))p Fs(.)148 4288 y Fx(\(4\))42 b Fv(R)417 4260 y Fq(\030)418 4292 y Fx(=)522 4288 y(End)697 4303 y Fr(S)748 4288 y Fx(\()p Fv(M)5 b(:)p Fx(\))p Fs(.)0 4508 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))g(The)i(claim)f(follo)m(ws)g(from)g(the)g(fact)f(that)g Fv(R)q Fx(-Mo)s(d)2456 4481 y Fq(\030)2457 4513 y Fx(=)2577 4508 y Fv(D)s Fx(-)o(Mo)s(d)g(and)h(since)h(eac)m(h)g(\014nitely)0 4625 y(generated)33 b Fv(D)s Fx(-mo)s(dule)g(is)g(a)f(progenerator.)0 4741 y(\(2\))44 b Fv(S)6 b Fx(-)o(Mo)s(d)509 4713 y Fq(\030)510 4745 y Fx(=)634 4741 y Fv(R)q Fx(-Mo)s(d)984 4713 y Fq(\030)985 4745 y Fx(=)1109 4741 y Fv(D)s Fx(-)o(Mo)s(d)44 b(implies)i(that)d Fq(V)8 b Fx(\()2136 4756 y Fr(S)2187 4741 y Fv(S)e Fx(\))2338 4713 y Fq(\030)2339 4745 y Fx(=)2463 4741 y Fq(V)i Fx(\()2570 4756 y Fr(D)2634 4741 y Fv(P)14 b Fx(\))44 b(is)g(Artinian.)79 b(F)-8 b(urthermore)0 4857 y Fq(V)8 b Fx(\()107 4872 y Fr(S)158 4857 y Fv(S)218 4872 y Fr(S)269 4857 y Fx(\))334 4830 y Fq(\030)335 4861 y Fx(=)440 4857 y Fq(V)g Fx(\()547 4872 y Fr(D)611 4857 y Fv(D)692 4872 y Fr(D)756 4857 y Fx(\),)32 b(hence)i Fv(S)39 b Fx(is)33 b(a)f(simple)i(ring.)0 4973 y(\(3\)+\(4\))e(follo)m(w)g(from)h(the)g(Morita)f(theorems.)2052 b Fo(\003)0 5310 y Fx(8.2.)49 b Ft(Injectiv)m(e)37 b(Mo)s(dules.)0 5530 y(De\014nition)45 b(and)g(Remark)g(8.9.)g Fx(An)38 b Fv(R)q Fx(-mo)s(dule)2027 5545 y Fr(R)2085 5530 y Fv(J)47 b Fx(is)39 b(called)h Fs(inje)-5 b(ctive)p Fx(,)39 b(if)f(for)g(eac)m (h)h(monomor-)0 5646 y(phism)d Fv(f)43 b Fx(:)33 b Fv(M)43 b Fq(\000)-60 b(!)32 b Fv(N)46 b Fx(and)35 b(for)g(eac)m(h)h (homomorphism)h Fv(g)e Fx(:)e Fv(M)43 b Fq(\000)-60 b(!)32 b Fv(J)45 b Fx(there)36 b(exists)h(a)e(homomorphism)p eop end %%Page: 68 68 TeXDict begin 68 67 bop 0 -170 a Fu(68)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fv(h)i Fx(:)g Fv(N)38 b Fq(\000)-60 b(!)28 b Fv(J)41 b Fx(with)33 b Fv(hf)39 b Fx(=)27 b Fv(g)1663 252 y(M)401 b(N)p 1796 219 334 4 v 2047 217 a Fj(-)1933 179 y Fv(f)1684 739 y(J)p 1713 637 4 351 v 1715 637 a Fj(?)1626 473 y Fv(g)2019 496 y(h)2051 369 y Fj(\000)1968 452 y(\000)1885 535 y(\000)1802 618 y(\000)1783 637 y(\000)-83 b(\011)2246 705 y Fv(:)0 895 y Fx(V)-8 b(ector)42 b(spaces)h(are)f(injectiv)m(e.)1251 910 y Fm(Z)1300 895 y Fn(Z)g Fx(is)g(not)g(injectiv)m(e.)72 b(The)43 b(injectiv)m(e)h Fn(Z)p Fx(-mo)s(dules)f(are)e(exactly)j(the)0 1012 y(divisible)35 b(Ab)s(elian)e(groups.)1098 1027 y Fm(Z)1147 1012 y Fn(Q)g Fx(is)g(injectiv)m(e.)0 1216 y Ft(Theorem)38 b(8.10.)k Fs(\(The)34 b(Baer)g(criterion\):)44 b(The)35 b(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent)34 b(for)h Fv(Q)28 b Fq(2)g Fv(R)q Fs(-)p Fx(Mo)s(d)p Fs(:)148 1380 y Fx(\(1\))42 b Fv(Q)35 b Fs(is)f(inje)-5 b(ctive.)148 1496 y Fx(\(2\))42 b Fq(8)370 1511 y Fr(R)428 1496 y Fv(I)35 b Fq(\022)611 1511 y Fr(R)669 1496 y Fv(R)q(;)52 b Fq(8)p Fv(g)31 b Fx(:)d Fv(I)35 b Fq(\000)-57 b(!)27 b Fv(Q)36 b Fq(9)p Fv(h)28 b Fx(:)g Fv(R)g Fq(\000)-57 b(!)28 b Fv(Q)35 b Fs(with)f Fv(h\023)29 b Fx(=)e Fv(g)1693 1718 y(I)432 b(R)p 1773 1685 367 4 v 2057 1683 a Fj(-)1939 1665 y Fv(\023)1680 2196 y(Q)p 1717 2103 4 351 v 1719 2103 a Fj(?)1629 1939 y Fv(g)2023 1962 y(h)2055 1835 y Fj(\000)1972 1918 y(\000)1889 2001 y(\000)1806 2084 y(\000)1787 2103 y(\000)-83 b(\011)2243 2171 y Fv(:)148 2433 y Fx(\(3\))42 b Fs(Each)c(monomorphism)e Fv(f)46 b Fx(:)35 b Fv(Q)1563 2373 y Fr(f)1524 2433 y Fq(\000)-57 b(!)35 b Fv(M)49 b Fs(splits,)40 b(i.e.)56 b(ther)-5 b(e)38 b(is)h(an)f(epimorphism)f Fv(g)i Fx(:)c Fv(M)45 b Fq(\000)-57 b(!)35 b Fv(Q)315 2549 y Fs(with)f Fv(g)t(f)k Fx(=)27 b(1)815 2564 y Fr(Q)875 2549 y Fs(.)0 2754 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))32 b(=)-17 b Fq(\))33 b Fx(\(2\):)43 b(follo)m(ws)33 b(immediately)i(from)d(the)h(de\014nition.)0 2870 y(\(1\))f(=)-17 b Fq(\))33 b Fx(\(3\):)43 b(The)33 b(diagram)1715 3089 y Fv(Q)397 b(M)p 1821 3066 340 4 v 2078 3064 a Fj(-)1962 3026 y Fv(f)1715 3576 y(Q)p 1752 3484 4 351 v 1754 3484 a Fj(?)1607 3326 y Fx(1)1656 3341 y Fr(Q)2058 3320 y Fv(g)2090 3216 y Fj(\000)2007 3299 y(\000)1924 3382 y(\000)1841 3465 y(\000)1822 3484 y(\000)-83 b(\011)0 3746 y Fx(de\014nes)34 b(the)f(required)h Fv(g)t Fx(.)0 3862 y(\(3\))e(=)-17 b Fq(\))33 b Fx(\(1\):)43 b(In)33 b(the)g(diagram)1646 4090 y Fv(M)401 b(N)p 1779 4058 334 4 v 2030 4056 a Fj(-)1916 4018 y Fv(f)1659 4568 y(Q)411 b(P)p 1765 4516 353 4 v 2035 4514 a Fj(-)1910 4476 y Fv(')p 1765 4574 V 1765 4572 a Fj(\033)1917 4634 y Fv(\032)p 1696 4475 4 351 v 1698 4475 a Fj(?)1609 4312 y Fv(g)p 2184 4475 V 2185 4475 a Fj(?)2224 4325 y Fv( )0 4810 y Fx(assume)35 b(that)e Fv(f)44 b Fx(is)33 b(a)g(monomorphism)i (and)e Fv(P)42 b Fx(:=)29 b Fv(N)k Fq(\010)23 b Fv(Q=)p Fq(f)p Fx(\()p Fv(f)11 b Fx(\()p Fv(m)p Fx(\))p Fv(;)17 b Fq(\000)p Fv(g)t Fx(\()p Fv(m)p Fx(\)\))p Fq(j)p Fv(m)28 b Fq(2)h Fv(M)10 b Fq(g)34 b Fx(with)g Fv(')f Fx(resp.)0 4937 y Fv( )39 b Fx(are)d(canonical)f(maps)h(to)f(the)h(left)f(resp.)53 b(the)36 b(righ)m(t)f(comp)s(onen)m(ts:)51 b Fv(')p Fx(\()p Fv(q)t Fx(\))32 b(:=)p 3019 4851 216 4 v 32 w(\(0)p Fv(;)17 b(q)t Fx(\))o Fv(;)g( )t Fx(\()p Fv(n)p Fx(\))32 b(:=)p 3646 4851 227 4 v 32 w(\()p Fv(n;)17 b Fx(0\).)0 5065 y(Since)39 b Fv( )t(f)11 b Fx(\()p Fv(m)p Fx(\))38 b(=)p 698 4979 389 4 v 37 w(\()p Fv(f)11 b Fx(\()p Fv(m)p Fx(\))p Fv(;)17 b Fx(0\))37 b(=)p 1237 4979 380 4 v 37 w(\(0)p Fv(;)17 b(g)t Fx(\()p Fv(m)p Fx(\)\))37 b(=)g Fv('g)t Fx(\()p Fv(m)p Fx(\))h(w)m(e)h(ha)m(v)m(e)h Fv( )t(f)48 b Fx(=)37 b Fv('g)t Fx(.)60 b(Let)39 b Fv(')p Fx(\()p Fv(q)t Fx(\))e(=)p 3458 4979 216 4 v 37 w(\(0)p Fv(;)17 b(q)t Fx(\))37 b(=)g(0.)0 5182 y(Then)d(there)g(exists)i(an)d Fv(m)c Fq(2)g Fv(M)44 b Fx(with)34 b Fv(f)11 b Fx(\()p Fv(m)p Fx(\))29 b(=)g(0)k(and)h Fv(g)t Fx(\()p Fv(m)p Fx(\))28 b(=)h Fv(q)t Fx(.)45 b(Since)35 b Fv(f)44 b Fx(is)34 b(an)f(injectiv)m(e)i(map,)f(w)m(e)0 5298 y(ha)m(v)m(e)h Fv(m)c Fx(=)f(0)k(and)g(th)m(us)h Fv(')f Fx(injectiv)m(e.)50 b(By)34 b(\(3\))g(there)h(is)f(a)g Fv(\032)g Fx(with)h Fv(\032')30 b Fx(=)g(1)2819 5313 y Fr(Q)2879 5298 y Fx(.)47 b(Then)36 b Fv(\032 )t(f)41 b Fx(=)30 b Fv(\032'g)k Fx(=)c Fv(g)t Fx(,)0 5414 y(and)j(th)m(us)g Fv(Q)g Fx(is)g(injectiv)m(e.)0 5530 y(\(2\))45 b(=)-17 b Fq(\))44 b Fx(\(1\):)68 b(Giv)m(en)46 b(a)f(monomorphism)h Fv(f)60 b Fx(:)49 b Fv(M)60 b Fq(\000)-60 b(!)49 b Fv(N)55 b Fx(and)45 b(a)g(homomorphism)h Fv(g)52 b Fx(:)d Fv(N)60 b Fq(\000)-60 b(!)49 b Fv(Q)p Fx(.)0 5646 y(Consider)29 b(the)f(set)g Fq(S)36 b Fx(:=)28 b Fq(f)p Fx(\()p Fv(N)1105 5661 y Fr(i)1133 5646 y Fv(;)17 b(')1241 5661 y Fr(i)1268 5646 y Fx(\))p Fq(g)p Fx(,)29 b(where)g Fv(N)1767 5661 y Fr(i)1823 5646 y Fq(\022)f Fv(N)38 b Fx(is)28 b(a)f(submo)s(dule)i(with)g(Im\()p Fv(f)11 b Fx(\))27 b Fq(\022)i Fv(N)3379 5661 y Fr(i)3434 5646 y Fx(and)f Fv(')3683 5661 y Fr(i)3739 5646 y Fx(:)g Fv(N)3872 5661 y Fr(i)p eop end %%Page: 69 69 TeXDict begin 69 68 bop 1239 -170 a Fu(Simple)26 b(and)f(Semisimple)h (rings)h(and)e(Mo)r(dules)1163 b(69)0 29 y Fq(\000)-60 b(!)28 b Fv(Q)33 b Fx(is)g(a)f(homomorphism)i(suc)m(h)g(that)1433 253 y Fv(M)392 b(N)1997 268 y Fr(i)p 1566 228 325 4 v 1808 226 a Fj(-)1698 188 y Fv(f)2416 253 y(N)p 2054 228 333 4 v 2304 226 a Fj(-)1446 738 y Fv(Q)p 1483 645 4 351 v 1485 645 a Fj(?)1396 482 y Fv(g)346 b(')1853 497 y Fr(i)1821 378 y Fj(\000)1738 461 y(\000)1655 544 y(\000)1572 627 y(\000)1553 645 y(\000)-83 b(\011)0 906 y Fx(comm)m(utes.)44 b(W)-8 b(e)28 b(ha)m(v)m(e)i Fq(S)35 b(6)p Fx(=)28 b Fq(;)p Fx(,)h(since)g(\(Im\()p Fv(f)11 b Fx(\))p Fv(;)17 b(g)t(f)1861 870 y Fp(\000)p Fk(1)1954 906 y Fx(\))28 b Fq(2)g Fv(S)6 b Fx(.)41 b(F)-8 b(urthermore)29 b Fq(S)36 b Fx(is)28 b(ordered)h(b)m(y)g(\()p Fv(N)3593 921 y Fr(i)3621 906 y Fv(;)17 b(')3729 921 y Fr(i)3757 906 y Fx(\))28 b Fq(\024)0 1022 y Fx(\()p Fv(N)116 1037 y Fr(j)153 1022 y Fv(;)17 b(')261 1037 y Fr(j)297 1022 y Fx(\))36 b(if)g Fv(N)542 1037 y Fr(i)605 1022 y Fq(\022)e Fv(N)794 1037 y Fr(j)867 1022 y Fx(and)j Fv(')1125 1037 y Fr(j)1161 1022 y Fq(j)1189 1037 y Fr(N)1245 1047 y Fi(i)1309 1022 y Fx(=)d Fv(')1483 1037 y Fr(i)1511 1022 y Fx(.)55 b(Let)37 b Fq(f)p Fx(\()p Fv(N)1938 1037 y Fr(i)1966 1022 y Fv(;)17 b(')2074 1037 y Fr(i)2102 1022 y Fx(\))p Fq(j)p Fv(i)34 b Fq(2)g Fv(J)9 b Fq(g)36 b Fx(b)s(e)h(a)f(c)m(hain)h(in)g Fq(S)7 b Fx(.)56 b(Then)37 b Fq([)p Fv(N)3637 1037 y Fr(i)3700 1022 y Fq(\022)d Fv(N)0 1138 y Fx(is)40 b(a)f(submo)s(dule.) 65 b Fv( )44 b Fx(:)39 b Fq([)p Fv(N)1059 1153 y Fr(i)1127 1138 y Fq(\000)-59 b(!)39 b Fv(Q)g Fx(with)h Fv( )t Fx(\()p Fv(n)1792 1153 y Fr(i)1821 1138 y Fx(\))f(=)g Fv(')2077 1153 y Fr(i)2105 1138 y Fx(\()p Fv(n)2201 1153 y Fr(i)2229 1138 y Fx(\))h(is)g(a)f(w)m(ell)h(de\014ned)h(homomorphism)g(and)0 1254 y(\()p Fq([)p Fv(N)182 1269 y Fr(i)211 1254 y Fv(;)17 b( )t Fx(\))36 b Fq(2)i(S)7 b Fx(.)61 b(F)-8 b(urthermore)38 b(w)m(e)i(ha)m(v)m(e)f(\()p Fv(N)1721 1269 y Fr(j)1758 1254 y Fv(;)17 b(')1866 1269 y Fr(j)1902 1254 y Fx(\))37 b Fq(\024)h Fx(\()p Fq([)p Fv(N)2274 1269 y Fr(i)2302 1254 y Fv(;)17 b( )t Fx(\))38 b(for)f(all)i Fv(j)k Fq(2)37 b Fv(J)9 b Fx(.)61 b(By)38 b(Zorn's)g(Lemma)0 1371 y(there)23 b(exists)h(a)e(maximal)h(elemen)m(t)i(\()p Fv(N)1434 1335 y Fp(0)1457 1371 y Fv(;)17 b(')1565 1335 y Fp(0)1588 1371 y Fx(\))22 b(in)h Fq(S)7 b Fx(.)41 b(W)-8 b(e)23 b(sho)m(w)g(that)f Fv(N)2565 1335 y Fp(0)2617 1371 y Fx(=)27 b Fv(N)10 b Fx(,)25 b(for)d(then)h(the)g(con)m(tin)m(uation)0 1487 y(of)33 b Fv(g)k Fx(to)c Fv(N)44 b Fx(exists.)49 b(Let)34 b Fv(x)c Fq(2)g Fv(N)j Fq(n)23 b Fv(N)1378 1451 y Fp(0)1402 1487 y Fx(.)46 b(Then)35 b Fv(N)1819 1451 y Fp(0)1872 1487 y Fn($)30 b Fv(N)2067 1451 y Fp(0)2114 1487 y Fx(+)23 b Fv(R)q(x)p Fx(.)47 b(Let)34 b Fv(I)j Fx(:=)30 b Fq(f)p Fv(r)i Fq(2)e Fv(R)q Fq(j)p Fv(r)s(x)f Fq(2)h Fv(N)3446 1451 y Fp(0)3470 1487 y Fq(g)p Fx(.)46 b(Then)35 b Fv(I)0 1603 y Fx(is)e(an)g(ideal)g(and)f(w)m(e)i(ha)m(v)m (e)g(a)e(comm)m(utativ)m(e)j(diagram)1860 1787 y Fv(I)433 b(R)p 1940 1755 367 4 v 2224 1753 a Fj(-)2106 1735 y Fv(\023)1346 2277 y(M)390 b(N)1918 2241 y Fp(0)p 1479 2242 322 4 v 1718 2240 a Fj(-)1611 2202 y Fv(f)2192 2273 y(N)2280 2237 y Fp(0)2326 2273 y Fx(+)22 b Fv(R)q(x)p 1970 2242 194 4 v 2081 2240 a Fj(-)1531 2496 y Fv(g)1466 2392 y Fj(@)1549 2475 y(@)1632 2558 y(@)1715 2641 y(@)1734 2660 y(@)-83 b(R)1847 2752 y Fv(Q)p 1884 2660 4 351 v 1886 2660 a Fj(?)1759 2511 y Fv(')1823 2475 y Fp(0)2297 1904 y Fj(\001)2256 1987 y(\001)2214 2070 y(\001)2173 2153 y(\001)2131 2237 y(\001)2090 2320 y(\001)2048 2403 y(\001)2007 2486 y(\001)1965 2569 y(\001)1924 2652 y(\001)1920 2660 y(\001)-42 b(\013)2133 2018 y Fv(\033)p 1884 2172 V 1886 2172 a Fj(?)1925 2019 y Fq(\001)p Fv(x)p 2371 2172 V 2373 2172 a Fj(?)2412 2009 y Fv(\032)2190 2506 y(\034)2222 2392 y Fj(\000)2139 2475 y(\000)2056 2558 y(\000)1973 2641 y(\000)1954 2660 y(\000)-83 b(\011)0 2916 y Fx(with)37 b Fv(\032)p Fx(\()p Fv(r)s Fx(\))d(:=)h Fv(r)27 b Fq(\001)e Fv(x)p Fx(.)56 b(Then)37 b(w)m(e)h(ha)m(v)m(e)g Fv(\032)p Fx(\()p Fv(I)8 b Fx(\))34 b Fq(\022)h Fv(N)1879 2880 y Fp(0)1903 2916 y Fx(.)55 b(Th)m(us)38 b(b)m(y)g(\(2\))e(there)h (is)g(a)f(homomorphism)i Fv(\033)g Fx(:)d Fv(R)0 3032 y Fq(\000)-60 b(!)28 b Fv(Q)h Fx(with)h Fv(\033)t(\023)f Fx(=)e Fv(')759 2996 y Fp(0)798 3032 y Fq(\016)16 b Fx(\()p 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Fv(r)25 b Fq(\000)f Fv(r)560 3280 y Fk(1)600 3265 y Fx(\))30 b(=)g Fv(')838 3228 y Fp(0)861 3265 y Fx(\(\()p Fv(r)c Fq(\000)d Fv(r)1151 3280 y Fk(1)1191 3265 y Fx(\))p Fv(x)p Fx(\))30 b(=)h Fv(')1523 3228 y Fp(0)1546 3265 y Fx(\()p Fv(n)1642 3228 y Fp(0)1642 3289 y Fk(1)1705 3265 y Fq(\000)23 b Fv(n)1863 3228 y Fp(0)1887 3265 y Fx(\))34 b(and)g Fv(')2214 3228 y Fp(0)2237 3265 y Fx(\()p Fv(n)2333 3228 y Fp(0)2357 3265 y Fx(\))23 b(+)g Fv(\033)t Fx(\()p Fv(r)s Fx(\))30 b(=)g Fv(')2899 3228 y Fp(0)2922 3265 y Fx(\()p Fv(n)3018 3228 y Fp(0)3018 3289 y Fk(1)3058 3265 y Fx(\))23 b(+)g Fv(\033)t Fx(\()p Fv(r)3359 3280 y Fk(1)3398 3265 y Fx(\).)48 b(It)34 b(is)h(easy)0 3381 y(to)e(see)h(that)e Fv(\034)45 b Fx(is)33 b(also)g(a)g (homomorphism.)46 b(Since)34 b Fv(\034)11 b Fq(j)2016 3396 y Fr(N)2079 3377 y Fh(0)2134 3381 y Fx(=)28 b Fv(')2302 3345 y Fp(0)2358 3381 y Fx(holds)34 b(w)m(e)g(ha)m(v)m(e)g(\()p Fv(N)3109 3345 y Fp(0)3155 3381 y Fx(+)22 b Fv(R)q(x;)17 b(\034)11 b Fx(\))29 b Fq(2)g(S)41 b Fx(and)0 3497 y(\()p Fv(N)126 3461 y Fp(0)150 3497 y Fv(;)17 b(')258 3461 y Fp(0)281 3497 y Fx(\))27 b Fn(\010)h Fx(\()p Fv(N)577 3461 y Fp(0)623 3497 y Fx(+)22 b Fv(R)q(x;)17 b(\034)11 b Fx(\))33 b(a)g(con)m(tradiction)g(to)f(the)h(maximalit)m(y)i(of)d(\() p Fv(N)2731 3461 y Fp(0)2754 3497 y Fv(;)17 b(')2862 3461 y Fp(0)2885 3497 y Fx(\).)44 b(Th)m(us)34 b Fv(N)3329 3461 y Fp(0)3380 3497 y Fx(=)28 b Fv(N)10 b Fx(.)224 b Fo(\003)0 3684 y Ft(Corollary)39 b(8.11.)j Fs(If)35 b Fv(R)h Fs(is)e(a)h(semisimple)e(ring)h(then)h(e)-5 b(ach)34 b Fv(R)q Fs(-mo)-5 b(dule)35 b(is)f(pr)-5 b(oje)g(ctive)34 b(and)h(inje)-5 b(ctive.)0 3870 y(Pr)g(o)g(of.)41 b Fx(Let)36 b Fv(Q)g Fx(b)s(e)f(an)h Fv(R)q Fx(-mo)s(dule.)52 b(By)36 b(8.4)g(eac)m(h)g(ideal)g(is)g(a)f(direct)i(summand)g(of)e Fv(R)q Fx(.)52 b(The)37 b(follo)m(wing)0 3986 y(diagram)32 b(together)h(with)g(the)g(Baer)g(criterion)g(sho)m(ws)h(that)f Fv(Q)g Fx(is)g(injectiv)m(e:)1695 4172 y Fv(I)432 b(R)p 1775 4111 367 4 v 2059 4109 a Fj(-)p 1775 4169 V 1775 4167 a(\033)1682 4650 y Fv(Q:)p 1719 4557 4 351 v 1720 4557 a Fj(?)2057 4289 y(\000)1974 4372 y(\000)1891 4455 y(\000)1808 4538 y(\000)1789 4557 y(\000)-83 b(\011)0 4813 y Fx(Let)41 b Fv(f)52 b Fx(:)41 b Fv(N)52 b Fq(\000)-60 b(!)41 b Fv(P)54 b Fx(b)s(e)41 b(surjectiv)m(e.)69 b(Since)42 b(Ke\()p Fv(f)11 b Fx(\))41 b Fq(\022)h Fv(N)51 b Fx(is)41 b(a)f(submo)s(dule)i(and)e(injectiv)m(e)j(there)e(is)g(a)0 4930 y Fv(g)33 b Fx(:)c Fv(N)40 b Fq(\000)-60 b(!)29 b Fx(Ke)q(\()p Fv(f)11 b Fx(\))33 b(with)h Fv(g)t Fx(\()p Fv(n)p Fx(\))29 b(=)g Fv(n)34 b Fx(for)f(all)g Fv(n)d Fq(2)f Fx(Ke)q(\()p Fv(f)11 b Fx(\).)46 b(W)-8 b(e)33 b(de\014ne)i Fv(k)e Fx(:)c Fv(P)43 b Fq(\000)-60 b(!)29 b Fv(N)44 b Fx(b)m(y)35 b Fv(k)s Fx(\()p Fv(p)p Fx(\))29 b(=)g Fv(n)23 b Fq(\000)h Fv(g)t Fx(\()p Fv(n)p Fx(\))0 5046 y(for)36 b Fv(n)f Fq(2)g Fv(N)47 b Fx(with)37 b Fv(f)11 b Fx(\()p Fv(n)p Fx(\))35 b(=)f Fv(p)p Fx(.)56 b(If)37 b(also)f Fv(f)11 b Fx(\()p Fv(n)1624 5010 y Fp(0)1648 5046 y Fx(\))34 b(=)h Fv(p)h Fx(then)h Fv(f)11 b Fx(\()p Fv(n)25 b Fq(\000)h Fv(n)2483 5010 y Fp(0)2506 5046 y Fx(\))35 b(=)f(0)j(hence)h Fv(n)25 b Fq(\000)g Fv(n)3293 5010 y Fp(0)3351 5046 y Fq(2)35 b Fx(Ke)q(\()p Fv(f)11 b Fx(\))36 b(and)0 5162 y Fv(g)t Fx(\()p Fv(n)6 b Fq(\000)g Fv(n)294 5126 y Fp(0)317 5162 y Fx(\))27 b(=)h Fv(n)6 b Fq(\000)g Fv(n)691 5126 y Fp(0)715 5162 y Fx(.)40 b(This)26 b(implies)g Fv(n)6 b Fq(\000)g Fv(g)t Fx(\()p Fv(n)p Fx(\))27 b(=)h Fv(n)1841 5126 y Fp(0)1870 5162 y Fq(\000)6 b Fv(g)t Fx(\()p Fv(n)2100 5126 y Fp(0)2123 5162 y Fx(\).)41 b(So)24 b Fv(k)k Fx(is)d(a)f(w)m(ell)i(de\014ned)g(map.)41 b(F)-8 b(urthermore)0 5278 y Fv(f)11 b(k)s Fx(\()p Fv(p)p Fx(\))31 b(=)f Fv(f)11 b Fx(\()p Fv(n)24 b Fq(\000)g Fv(g)t Fx(\()p Fv(n)p Fx(\)\))30 b(=)h Fv(f)11 b Fx(\()p Fv(n)p Fx(\))23 b Fq(\000)h Fv(f)11 b(g)t Fx(\()p Fv(n)p Fx(\))30 b(=)h Fv(p)23 b Fq(\000)h Fx(0,)35 b(hence)h Fv(f)11 b(k)34 b Fx(=)c(1)2569 5293 y Fr(P)2628 5278 y Fx(.)49 b(In)35 b(order)g(to)f(sho)m(w)h(that)g Fv(k)i 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Fp(0)2797 5511 y Fx(\)\))27 b(=)h Fv(r)s(k)s Fx(\()p Fv(p)p Fx(\))18 b(+)h Fv(r)3390 5475 y Fp(0)3413 5511 y Fv(k)s Fx(\()p Fv(p)3554 5475 y Fp(0)3578 5511 y Fx(\).)42 b(Th)m(us)0 5627 y Fv(P)k Fx(is)33 b(pro)5 b(jectiv)m(e.)3171 b Fo(\003)p eop end %%Page: 70 70 TeXDict begin 70 69 bop 0 -170 a Fu(70)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 63 y Ft(Lemma)50 b(8.12.)d Fs(L)-5 b(et)44 b Fx(0)f Fq(\000)-57 b(!)44 b Fv(M)1306 3 y Fr(f)1267 63 y Fq(\000)-58 b(!)44 b Fv(N)1604 3 y Fr(g)1562 63 y Fq(\000)-57 b(!)44 b Fv(P)57 b Fq(\000)-57 b(!)44 b Fx(0)f Fs(b)-5 b(e)43 b(a)h(short)f(exact)h(se)-5 b(quenc)g(e.)70 b Fv(M)54 b Fs(and)43 b Fv(P)57 b Fs(ar)-5 b(e)0 179 y(A)n(rtinian)36 b(if)g(and)g(only)g(if)h Fv(N)47 b Fs(is)36 b(A)n(rtinian.)49 b(In)35 b(p)-5 b(articular)37 b(if)f Fv(M)47 b Fs(and)36 b Fv(N)47 b Fs(ar)-5 b(e)36 b(A)n(rtinian)g(then)h Fv(M)d Fq(\010)23 b Fv(N)0 296 y Fs(is)35 b(A)n(rtinian.)0 477 y(Pr)-5 b(o)g(of.)41 b Fx(Let)34 b Fv(N)44 b Fx(b)s(e)34 b(Artinian.)47 b(This)35 b(implies)g(immediately)g(that)f Fv(M)44 b Fx(is)34 b(Artinian.)47 b(If)34 b Fq(f)p Fv(L)3375 492 y Fr(i)3403 477 y Fq(g)g Fx(is)g(a)f(set)h(of)0 593 y(submo)s(dules)41 b(of)d Fv(P)52 b Fx(then)40 b Fq(f)p Fv(g)1098 557 y Fp(\000)p Fk(1)1191 593 y Fx(\()p Fv(L)1295 608 y Fr(i)1323 593 y Fx(\))p Fq(g)f Fx(is)g(a)g(set)g(of)g(submo)s(dules)h(of)f Fv(N)10 b Fx(.)62 b(Let)39 b Fv(g)2980 557 y Fp(\000)p Fk(1)3074 593 y Fx(\()p Fv(L)3178 608 y Fk(0)3217 593 y Fx(\))g(b)s(e)g(minimal)h(in)0 710 y(this)33 b(set.)44 b(Since)34 b Fv(g)t(g)737 674 y Fp(\000)p Fk(1)830 710 y Fx(\()p Fv(L)934 725 y Fr(i)963 710 y Fx(\))27 b(=)h Fv(L)1198 725 y Fr(i)1259 710 y Fx(w)m(e)33 b(ha)m(v)m(e)h(that)f Fv(L)1905 725 y Fk(0)1977 710 y Fx(is)g(minimal)g(in)g Fq(f)p Fv(L)2684 725 y Fr(i)2713 710 y Fq(g)p Fx(.)0 826 y(Let)k Fv(M)48 b Fx(and)37 b Fv(P)50 b Fx(b)s(e)38 b(Artinian.)57 b(Let)37 b Fq(f)p Fv(L)1505 841 y Fr(i)1533 826 y Fq(g)g Fx(b)s(e)g(a)g(set)h(of)e(submo)s(dules)j(of)e Fv(N)10 b Fx(.)57 b(Let)37 b Fv(L)3182 841 y Fk(0)3259 826 y Fx(b)s(e)g(c)m(hosen)h(suc)m(h)0 942 y(that)f Fv(g)t Fx(\()p Fv(L)371 957 y Fk(0)410 942 y Fx(\))g(is)g(minimal)h(in)g(the)f (set)h Fq(f)p Fv(g)t Fx(\()p Fv(L)1624 957 y Fr(i)1651 942 y Fx(\))p Fq(g)p Fx(.)57 b(Let)37 b Fv(L)h Fx(b)s(e)f(c)m(hosen)i (suc)m(h)f(that)f Fv(f)3059 906 y Fp(\000)p Fk(1)3153 942 y Fx(\()p Fv(L)p Fx(\))g(is)h(minimal)g(in)0 1058 y(the)g(set)f Fq(f)p Fv(f)438 1022 y Fp(\000)p Fk(1)532 1058 y Fx(\()p Fv(L)636 1073 y Fr(j)673 1058 y Fx(\))p Fq(j)p Fv(L)805 1073 y Fr(j)877 1058 y Fq(2)f(f)p Fv(L)1095 1073 y Fr(i)1123 1058 y Fq(g)c Fx(and)h Fv(g)t Fx(\()p Fv(L)1550 1073 y Fr(j)1586 1058 y Fx(\))j(=)f Fv(g)t Fx(\()p Fv(L)1926 1073 y Fk(0)1965 1058 y Fx(\))p Fq(g)p Fx(.)57 b(W)-8 b(e)37 b(sho)m(w)h(that)f Fv(L)h Fx(is)f(minimal)h(in)g Fq(f)p Fv(L)3596 1073 y Fr(i)3624 1058 y Fq(g)p Fx(.)57 b(Let)0 1175 y Fv(L)66 1138 y Fp(0)121 1175 y Fq(2)32 b(f)p Fv(L)335 1190 y Fr(i)363 1175 y Fq(g)j Fx(with)g Fv(L)d Fq(\023)g Fv(L)945 1138 y Fp(0)969 1175 y Fx(.)50 b(Then)36 b Fv(g)t Fx(\()p Fv(L)1458 1190 y Fk(0)1497 1175 y Fx(\))31 b(=)h Fv(g)t Fx(\()p Fv(L)p Fx(\))f Fq(\023)h Fv(g)t Fx(\()p Fv(L)2162 1138 y Fp(0)2184 1175 y Fx(\),)k(hence)g Fv(g)t Fx(\()p Fv(L)2713 1138 y Fp(0)2736 1175 y Fx(\))31 b(=)h Fv(g)t Fx(\()p Fv(L)3068 1190 y Fk(0)3107 1175 y Fx(\).)50 b(F)-8 b(urthermore)35 b(w)m(e)0 1291 y(ha)m(v)m(e)f Fv(f)284 1255 y Fp(\000)p Fk(1)378 1291 y Fx(\()p Fv(L)p Fx(\))28 b Fq(\023)g Fv(f)712 1255 y Fp(\000)p Fk(1)806 1291 y Fx(\()p Fv(L)910 1255 y Fp(0)934 1291 y Fx(\),)k(hence)i Fv(L)28 b Fx(=)g Fv(L)1566 1255 y Fp(0)1589 1291 y Fx(.)2207 b Fo(\003)0 1491 y Fx(8.3.)49 b Ft(Simple)38 b(and)h(Semisimple)g(Mo)s (dules.)0 1673 y(Lemma)33 b(8.13.)38 b Fs(L)-5 b(et)30 b Fv(R)896 1688 y Fk(1)936 1673 y Fv(;)17 b(:)g(:)g(:)f(;)h(R)1229 1688 y Fr(n)1306 1673 y Fs(b)-5 b(e)29 b(semisimple)g(rings.)42 b(Then)30 b Fv(R)2521 1688 y Fk(1)2572 1673 y Fq(\002)11 b Fv(:)17 b(:)g(:)11 b Fq(\002)g Fv(R)2948 1688 y Fr(n)3027 1673 y Fs(is)29 b(a)h(semisimple)e(ring.)0 1855 y(Pr)-5 b(o)g(of.)41 b Fx(\(Only)i(for)e(the)h(case)h Fv(R)1212 1870 y Fk(1)1280 1855 y Fq(\002)29 b Fv(R)1460 1870 y Fk(2)1500 1855 y Fx(\))41 b(By)i(Lemma)f(8.12)f Fv(R)2388 1870 y Fk(1)2456 1855 y Fq(\002)29 b Fv(R)2636 1870 y Fk(2)2718 1855 y Fx(is)42 b(Artinian.)72 b(Let)42 b Fv(I)51 b Fq(\022)44 b Fv(R)f Fx(b)s(e)0 1971 y(nilp)s(oten)m(t.)59 b(F)-8 b(rom)36 b Fv(I)781 1935 y Fr(n)864 1971 y Fx(=)g(0)h(w)m(e)h (get)g(for)f(eac)m(h)h Fv(a)e Fq(2)g Fv(I)45 b Fx(the)38 b(equation)g(\()p Fv(R)q(a)p Fx(\))2811 1935 y Fr(n)2894 1971 y Fx(=)e(0.)58 b(F)-8 b(rom)36 b Fv(a)g Fx(=)g(\()p Fv(a)3688 1986 y Fk(1)3728 1971 y Fv(;)17 b(a)3823 1986 y Fk(2)3862 1971 y Fx(\))0 2087 y(follo)m(ws)38 b(0)f(=)f(\()p Fv(R)q(a)p Fx(\))725 2051 y Fr(n)808 2087 y Fx(=)h(\()p Fv(R)1033 2102 y Fk(1)1072 2087 y Fv(a)1123 2102 y Fk(1)1163 2087 y Fv(;)17 b(R)1281 2102 y Fk(2)1320 2087 y Fv(a)1371 2102 y Fk(2)1411 2087 y Fx(\))1449 2051 y Fr(n)1496 2087 y Fx(.)59 b(Hence)39 b Fv(R)1951 2102 y Fk(1)1990 2087 y Fv(a)2041 2102 y Fk(1)2117 2087 y Fx(=)e(0)g(and)h Fv(R)2585 2102 y Fk(2)2624 2087 y Fv(a)2675 2102 y Fk(2)2751 2087 y Fx(=)f(0,)h(i.e.)60 b Fv(R)q(a)36 b Fx(=)h(0)g(and)h(th)m(us)0 2203 y Fv(I)e Fx(=)27 b(0.)3565 b Fo(\003)0 2385 y Ft(Lemma)43 b(8.14.)g Fs(Each)38 b(pr)-5 b(op)g(er)37 b(submo)-5 b(dule)37 b Fv(N)49 b Fs(of)37 b(a)h(\014nitely)g(gener)-5 b(ate)g(d)37 b(mo)-5 b(dule)37 b Fv(M)49 b Fs(is)37 b(c)-5 b(ontaine)g(d)37 b(in)0 2501 y(a)e(maximal)e(submo)-5 b(dule)34 b(of)h Fv(M)10 b Fs(.)46 b(In)34 b(p)-5 b(articular)35 b Fv(M)45 b Fs(p)-5 b(ossesses)34 b(a)g(simple)g(quotient)h(mo)-5 b(dule.)0 2683 y(Pr)g(o)g(of.)41 b Fx(Let)35 b Fv(N)42 b Fn($)32 b Fv(M)45 b Fx(b)s(e)35 b(a)g(prop)s(er)f(submo)s(dule)i(of)f Fv(M)10 b Fx(.)50 b(Let)35 b Fq(M)g Fx(b)s(e)g(the)g(set)g(of)f(submo)s (dules)j Fv(U)45 b Fx(with)0 2799 y Fv(N)38 b Fq(\022)28 b Fv(U)39 b Fn($)28 b Fv(M)10 b Fx(.)43 b Fq(M)30 b Fx(is)g(ordered)h (b)m(y)f(inclusion.)44 b(Let)30 b(\()p Fv(U)2052 2814 y Fr(i)2081 2799 y Fx(\))g(b)s(e)g(a)f(c)m(hain)i(in)f Fq(M)f Fx(and)h Fv(U)3135 2763 y Fp(0)3187 2799 y Fx(:=)e Fq([)p Fv(U)3450 2814 y Fr(i)3478 2799 y Fx(.)43 b(Then)31 b Fv(U)3876 2763 y Fp(0)0 2915 y Fx(is)36 b(again)f(a)h(submo)s(dule)h (and)f Fv(N)43 b Fo(j)34 b Fv(U)1444 2879 y Fp(0)1468 2915 y Fx(.)52 b(If)36 b Fv(U)1724 2879 y Fp(0)1781 2915 y Fx(=)d Fv(M)46 b Fx(then)36 b(all)g(generating)g(elemen)m(ts)i Fv(m)3361 2930 y Fk(1)3401 2915 y Fv(;)17 b(:)g(:)g(:)f(;)h(m)3705 2930 y Fr(t)3770 2915 y Fx(are)0 3032 y(in)35 b Fv(U)192 2996 y Fp(0)216 3032 y Fx(,)g(hence)h(there)f(is)g(a)g(mo)s(dule)g Fv(U)1398 3047 y Fr(i)1461 3032 y Fx(with)g Fv(m)1770 3047 y Fk(1)1810 3032 y Fv(;)17 b(:)g(:)g(:)e(;)i(m)2113 3047 y Fr(t)2174 3032 y Fq(2)32 b Fv(U)2338 3047 y Fr(i)2366 3032 y Fx(.)49 b(Th)m(us)37 b Fv(U)2758 3047 y Fr(i)2817 3032 y Fx(=)31 b Fv(M)10 b Fx(.)50 b(This)36 b(is)f(imp)s(ossible.)0 3148 y(So)g Fv(U)214 3112 y Fp(0)270 3148 y Fq(6)p Fx(=)d Fv(M)45 b Fx(and)35 b(th)m(us)i(in)e Fq(M)p Fx(.)50 b(F)-8 b(urthermore)36 b Fv(U)1884 3112 y Fp(0)1943 3148 y Fx(is)f(an)g(upp)s (er)h(b)s(ound)f(of)g(\()p Fv(U)2984 3163 y Fr(i)3012 3148 y Fx(\).)51 b(By)35 b(Zorn's)g(Lemma)0 3264 y(there)e(is)g(a)g (maximal)g(submo)s(dule)h(of)e Fv(M)43 b Fx(\(in)33 b Fq(M)p Fx(\),)f(that)h(con)m(tains)g Fv(N)10 b Fx(.)1179 b Fo(\003)0 3446 y Ft(Lemma)39 b(8.15.)190 b Fx(\(1\))42 b Fs(If)d Fv(X)45 b Fq(\022)1331 3461 y Fm(Z)1380 3446 y Fn(Q)40 b Fs(is)g(a)g(set)g(of)g(gener)-5 b(ating)39 b(elements)g(of)g Fn(Q)i Fs(over)e Fn(Z)i Fs(and)e Fv(x)f Fq(2)f Fv(X)315 3562 y Fs(then)d Fv(X)c Fq(n)22 b(f)p Fv(x)p Fq(g)35 b Fs(is)g(also)f(a)h(set)g(of)f(gener)-5 b(ating)34 b(elements)g(of)h Fn(Q)p Fs(.)148 3678 y Fx(\(2\))315 3693 y Fm(Z)363 3678 y Fn(Q)g Fs(p)-5 b(ossesses)34 b(no)g(maximal)g (submo)-5 b(dules.)0 3860 y(Pr)g(o)g(of.)41 b Fx(\(1\))23 b(Let)h Fv(B)33 b Fx(=)28 b Fq(h)p Fv(X)12 b Fq(n)t(f)p Fv(x)p Fq(gi)p Fx(.)40 b(Then)24 b Fn(Q)29 b Fx(=)e 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Fv(M)50 b Fq(\000)-60 b(!)38 b Fv(L)p Fx(\))p Fv(:)h Fx(If)g(0)f Fq(6)p Fx(=)g Fv(J)48 b Fn($)39 b Fv(I)46 b Fx(then)40 b Fv(L)f Fn($)f Fv(L)27 b Fx(+)g Fv(J)47 b Fn($)39 b Fv(R)q(x)g Fx(in)g(con)m(tradiction)h(to)f Fv(L)g Fx(maximal)g(in)g Fv(R)q(x)p Fx(.)0 5646 y(Hence)34 b Fv(I)40 b Fx(is)33 b(simple)h(with)g Fv(I)h Fq(\022)28 b Fv(R)q(x)g Fq(\022)h Fv(N)10 b Fx(.)p eop end %%Page: 71 71 TeXDict begin 71 70 bop 1239 -170 a Fu(Simple)26 b(and)f(Semisimple)h (rings)h(and)e(Mo)r(dules)1163 b(71)0 29 y Fx(Let)38 b Fv(N)46 b Fx(:=)443 -45 y Fl(P)564 29 y Fv(I)607 44 y Fr(j)681 29 y Fx(b)s(e)38 b(the)g(sum)g(of)f(all)g(simple)i(submo)s (dules)g(of)e Fv(M)10 b Fx(.)59 b(Then)38 b Fv(M)47 b Fx(=)35 b Fv(N)h Fq(\010)26 b Fv(K)7 b Fx(.)58 b(If)38 b Fv(K)43 b Fq(6)p Fx(=)35 b(0)0 146 y(then)29 b Fv(K)36 b Fx(con)m(tains)29 b(a)f(simple)i(submo)s(dule)g Fv(I)36 b Fx(and)29 b(w)m(e)h(ha)m(v)m(e)f Fv(I)36 b Fq(\022)28 b Fv(N)c Fq(\\)14 b Fv(K)7 b Fx(.)43 b(Con)m(tradiction.)g(Th)m(us)30 b Fv(K)35 b Fx(=)27 b(0)0 262 y(and)33 b Fv(M)38 b Fx(=)426 187 y Fl(P)547 262 y Fv(I)590 277 y Fr(j)627 262 y Fx(.)3169 b Fo(\003)0 440 y Ft(Lemma)46 b(8.17.)f Fs(L)-5 b(et)852 455 y Fr(R)910 440 y Fv(M)51 b Fs(b)-5 b(e)40 b(a)g(sum)g(of)g(simple)g (submo)-5 b(dules:)54 b Fv(M)49 b Fx(=)2726 365 y Fl(P)2831 469 y Fr(j)t Fp(2)p Fr(X)2995 440 y Fv(I)3038 455 y Fr(j)3074 440 y Fs(.)61 b(L)-5 b(et)41 b Fv(N)48 b Fq(\022)38 b Fv(M)51 b Fs(b)-5 b(e)40 b(a)0 565 y(submo)-5 b(dule.)71 b(Then)43 b(ther)-5 b(e)44 b(is)f(a)h(set)g Fv(Y)65 b Fq(\022)45 b Fv(X)51 b Fs(with)44 b Fv(M)55 b Fx(=)44 b Fv(N)39 b Fq(\010)2507 490 y Fl(L)2618 594 y Fr(j)t Fp(2)p Fr(Y)2775 565 y Fv(I)2818 580 y Fr(j)2898 565 y Fs(and)k(a)h(set)g Fv(Z)51 b Fq(\022)45 b Fv(X)51 b Fs(with)0 690 y Fv(N)116 662 y Fq(\030)117 694 y Fx(=)221 615 y Fl(L)332 719 y Fr(j)t Fp(2)p Fr(Z)485 690 y Fv(I)528 705 y Fr(j)564 690 y Fs(.)45 b(In)34 b(p)-5 b(articular)35 b(e)-5 b(ach)34 b(submo)-5 b(dule)34 b Fv(N)46 b Fs(of)34 b Fv(M)46 b Fs(is)35 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b(at)e(most)h(\014nitely)h(man)m(y)f Fv(a)3088 1133 y Fr(j)3161 1118 y Fq(2)e Fv(I)3306 1133 y Fr(j)3380 1118 y Fx(are)i(di\013eren)m(t)0 1237 y(from)43 b(0.)76 b(Hence)45 b(there)f(is)g(a)f Fv(Z)1222 1252 y Fr(i)1293 1237 y Fx(in)h(the)g(c)m(hain)g(with)g Fv(j)53 b Fq(2)46 b Fv(Z)2370 1252 y Fr(i)2441 1237 y Fx(for)d(all)h Fv(a)2799 1252 y Fr(j)2882 1237 y Fq(6)p Fx(=)i(0)d(in)h(the)f(sum.)77 b(F)-8 b(rom)0 1354 y Fv(N)23 b Fx(+)12 b(\()227 1279 y Fl(P)332 1383 y Fr(j)t Fp(2)p Fr(Z)460 1393 y Fi(i)507 1354 y Fv(I)550 1369 y Fr(j)586 1354 y Fx(\))28 b(=)f Fv(N)c Fq(\010)12 b Fx(\()983 1279 y Fl(L)1095 1383 y Fr(j)t Fp(2)p Fr(Z)1223 1393 y Fi(i)1269 1354 y Fv(I)1312 1369 y Fr(j)1348 1354 y Fx(\))28 b(w)m(e)h(get)f Fv(n)g Fx(=)f(0)h(=)f Fv(a)2131 1369 y Fr(j)2196 1354 y Fx(for)g(all)h Fv(j)33 b Fq(2)28 b Fv(Z)2712 1317 y Fp(0)2736 1354 y Fx(.)41 b(By)29 b(Zorn's)f(Lemma)g(there)g(is)0 1479 y(a)j(maximal)h(elemen)m(t)h Fv(Z)910 1442 y Fp(00)980 1479 y Fq(2)28 b(S)7 b Fx(,)32 b(and)f(w)m(e)h(ha)m(v)m(e)h Fv(P)41 b Fx(:=)27 b Fv(N)j Fx(+)19 b(\()2229 1404 y Fl(P)2334 1508 y Fr(j)t Fp(2)p Fr(Z)2467 1489 y Fh(00)2527 1479 y Fv(I)2570 1494 y Fr(j)2607 1479 y Fx(\))28 b(=)f Fv(N)j Fq(\010)19 b Fx(\()3018 1404 y Fl(L)3129 1508 y Fr(j)t Fp(2)p Fr(Z)3262 1489 y Fh(00)3322 1479 y Fv(I)3365 1494 y Fr(j)3402 1479 y Fx(\).)43 b(Let)31 b Fv(I)3726 1494 y Fr(k)3800 1479 y Fx(b)s(e)0 1604 y(simple)f(with)f Fv(k)i Fq(2)d Fv(X)21 b Fq(n)14 b Fv(Z)934 1567 y Fp(00)976 1604 y Fx(.)42 b(If)28 b Fv(P)g Fx(+)14 b Fv(I)1362 1619 y Fr(k)1431 1604 y Fx(=)28 b Fv(P)f Fq(\010)14 b Fv(I)1759 1619 y Fr(k)1802 1604 y Fx(,)29 b(then)g Fv(N)24 b Fx(+)14 b(\()2306 1529 y Fl(P)2411 1633 y Fr(j)t Fp(2)p Fr(Z)2544 1614 y Fh(00)2604 1604 y Fv(I)2647 1619 y Fr(j)2684 1604 y Fx(\))g(+)g Fv(I)2869 1619 y Fr(k)2938 1604 y Fx(=)28 b Fv(N)c Fq(\010)14 b Fx(\()3273 1529 y Fl(L)3384 1633 y Fr(j)t Fp(2)p Fr(Z)3517 1614 y Fh(00)3577 1604 y Fv(I)3620 1619 y Fr(j)3657 1604 y Fx(\))g Fq(\010)g Fv(I)3843 1619 y Fr(R)0 1727 y Fx(in)35 b(con)m(tradiction)h(to)f(the)g(maximalit)m(y) i(of)e Fv(Z)1707 1691 y Fp(00)1749 1727 y Fx(.)51 b(Hence)36 b(0)c Fq(6)p Fx(=)f Fv(P)38 b Fq(\\)24 b Fv(I)2541 1742 y Fr(k)2616 1727 y Fq(\022)32 b Fv(I)2768 1742 y Fr(k)2811 1727 y Fx(,)j(or)g Fv(I)3038 1742 y Fr(k)3113 1727 y Fq(\022)d Fv(P)14 b Fx(.)50 b(This)36 b(implies)0 1843 y Fv(P)41 b Fx(=)28 b Fv(N)k Fx(+)416 1768 y Fl(P)521 1872 y Fr(j)t Fp(2)p Fr(X)685 1843 y Fv(I)728 1858 y Fr(j)792 1843 y Fx(=)c Fv(M)10 b Fx(.)0 1968 y(No)m(w)40 b(w)m(e)g(apply)f(the)h(\014rst)f(claim)h(to)1428 1893 y Fl(L)1538 1997 y Fr(j)t Fp(2)p Fr(Y)1695 1968 y Fv(I)1738 1983 y Fr(j)1814 1968 y Fx(and)f(obtain)g Fv(N)e Fq(\010)27 b Fx(\()2577 1893 y Fl(L)2688 1997 y Fr(j)t Fp(2)p Fr(Y)2844 1968 y Fv(I)2887 1983 y Fr(j)2924 1968 y Fx(\))39 b(=)f Fv(M)50 b Fx(=)38 b(\()3411 1893 y Fl(L)3522 1997 y Fr(j)t Fp(2)p Fr(Y)3678 1968 y Fv(I)3721 1983 y Fr(j)3758 1968 y Fx(\))27 b Fq(\010)0 2093 y Fx(\()38 2018 y Fl(L)149 2122 y Fr(j)t Fp(2)p Fr(Z)301 2093 y Fv(I)344 2108 y Fr(j)381 2093 y Fx(\).)43 b(This)34 b(implies)g Fv(N)1159 2065 y Fq(\030)1160 2097 y Fx(=)1264 2093 y Fv(M)5 b(=)p Fx(\()1450 2018 y Fl(L)1561 2122 y Fr(j)t Fp(2)p Fr(Y)1718 2093 y Fv(I)1761 2108 y Fr(j)1798 2093 y Fx(\))1863 2065 y Fq(\030)1864 2097 y Fx(=)1968 2018 y Fl(L)2079 2122 y Fr(j)t Fp(2)p Fr(Z)2232 2093 y Fv(I)2275 2108 y Fr(j)2311 2093 y Fx(.)1485 b Fo(\003)0 2280 y Ft(Theorem)45 b(8.18.)h Fs(\(Structur)-5 b(e)41 b(The)-5 b(or)g(em)40 b(for)h(Semisimple)e(Mo) -5 b(dules\):)56 b(F)-7 b(or)2979 2295 y Fr(R)3036 2280 y Fv(M)52 b Fs(the)41 b(fol)5 b(lowing)39 b(ar)-5 b(e)0 2396 y(e)g(quivalent)148 2536 y Fx(\(1\))42 b Fs(Each)34 b(submo)-5 b(dule)34 b(of)h Fv(M)45 b Fs(is)35 b(a)g(sum)f(of)h(simple) f(submo)-5 b(dules.)148 2652 y Fx(\(2\))42 b Fv(M)j Fs(is)35 b(a)g(sum)f(of)h(simple)f(submo)-5 b(dules.)148 2768 y Fx(\(3\))42 b Fv(M)j Fs(is)35 b(a)g(dir)-5 b(e)g(ct)34 b(sum)h(of)g(simple)e(submo)-5 b(dules.)148 2884 y Fx(\(4\))42 b 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b(e)g(ct)35 b(sum)f(of)h(simple)f(left)h(ide)-5 b(als.)148 1345 y Fx(\(7\))42 b Fv(R)417 1318 y Fq(\030)418 1349 y Fx(=)522 1345 y Fv(R)596 1360 y Fk(1)658 1345 y Fq(\002)23 b Fv(:)17 b(:)g(:)22 b Fq(\002)g Fv(R)1068 1360 y Fr(n)1150 1345 y Fs(with)35 b(simple)f(rings)g Fv(R)1985 1360 y Fr(i)2048 1345 y Fx(\()p Fv(i)28 b Fx(=)g(1)p Fv(;)17 b(:)g(:)g(:)e(;)i(n)p Fx(\))p Fs(.)148 1461 y Fx(\(8\))42 b Fv(R)417 1434 y Fq(\030)418 1466 y Fx(=)522 1461 y Fv(B)596 1476 y Fk(1)658 1461 y Fq(\010)23 b Fv(:)17 b(:)g(:)k Fq(\010)i Fv(B)1068 1476 y Fr(n)1115 1461 y Fs(,)35 b(wher)-5 b(e)34 b(the)h Fv(B)1691 1476 y Fr(i)1754 1461 y Fs(ar)-5 b(e)35 b(minimal)e(two)i(side)-5 b(d)34 b(ide)-5 b(als)34 b(and)3188 1476 y Fr(R)3245 1461 y Fv(R)i Fs(is)f(A)n(rtinian.)148 1578 y Fx(\(9\))42 b Fv(R)389 1593 y Fr(R)481 1578 y Fs(is)35 b(semisimple)e(\(as)i(a)f (ring\).)0 1792 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))32 b(=)-17 b Fq(\))33 b Fx(\(3\):)43 b(Corollary)33 b(8.11.)0 1908 y(\(3\))f(=)-17 b Fq(\))33 b Fx(\(4\):)43 b(Theorem)34 b(8.18)d(\(4\))i(and)f(Theorem)i(8.10)e(\(3\).)0 2025 y(\(4\))g(=)-17 b Fq(\))33 b Fx(\(5\):)43 b(Sp)s(ecialization.)0 2141 y(\(5\))32 b(=)-17 b Fq(\))33 b Fx(\(6\):)43 b(Theorem)34 b(8.18)d(\(3\).)0 2257 y(\(6\))h(=)-17 b Fq(\))33 b Fx(\(3\):)43 b(Theorem)34 b(8.18)d(\(4\))i(and)f(8.11.)0 2373 y(\(6\))g(=)-17 b Fq(\))33 b Fx(\(2\):)43 b(Theorem)34 b(8.18)d(\(4\))i(and)f(8.11.)0 2489 y(\(2\))38 b(=)-17 b Fq(\))38 b Fx(\(4\):)55 b(Let)38 b Fv(N)48 b Fq(\022)38 b Fv(M)49 b Fx(b)s(e)39 b(a)f(submo)s(dule.)62 b(Then)39 b Fv(M)5 b(=)-5 b(N)49 b Fx(is)39 b(pro)5 b(jectiv)m(e,)42 b(so)c(there)h(is)g Fv(f)48 b Fx(:)38 b Fv(M)5 b(=)-5 b(N)0 2606 y Fq(\000)-60 b(!)35 b Fv(M)48 b Fx(with)37 b(\()p Fv(M)5 b(=)-5 b(N)46 b Fq(\000)-60 b(!)35 b Fv(M)46 b Fq(\000)-60 b(!)35 b Fv(M)5 b(=)-5 b(N)10 b Fx(\))35 b(=)g(id)i(or)g(\()p Fv(M)46 b Fq(\000)-60 b(!)35 b Fv(M)5 b(=)-5 b(N)45 b Fq(\000)-59 b(!)34 b Fv(M)10 b Fx(\))36 b(=)f Fv(p)i Fx(with)h Fv(p)3325 2570 y Fk(2)3399 2606 y Fx(=)d Fv(p)p Fx(.)57 b(Hence)0 2722 y Fv(M)38 b Fx(=)28 b(Ke\()p Fv(p)p Fx(\))22 b Fq(\010)h Fx(Im\()p Fv(p)p Fx(\))33 b(and)f(Ke\()p Fv(p)p Fx(\))c(=)g Fv(N)10 b Fx(.)0 2838 y(\(6\))28 b(=)-17 b Fq(\))28 b Fx(\(8\):)41 b(Let)29 b Fv(R)g Fx(=)e Fv(I)953 2853 y Fk(11)1042 2838 y Fq(\010)14 b Fv(:)j(:)g(:)c Fq(\010)h Fv(I)1395 2853 y Fk(1)p Fr(i)1454 2862 y Fg(1)1507 2838 y Fq(\010)g Fv(I)1641 2853 y Fk(21)1730 2838 y Fq(\010)g Fv(:)j(:)g(:)d Fq(\010)g Fv(I)2084 2853 y Fk(2)p Fr(i)2143 2862 y Fg(2)2196 2838 y Fq(\010)g Fv(:)j(:)g(:)c Fq(\010)h Fv(I)2549 2853 y Fr(n)p Fk(1)2645 2838 y Fq(\010)g Fv(:)j(:)g(:)d Fq(\010)g Fv(I)2999 2853 y Fr(ni)3066 2861 y Fi(n)3141 2838 y Fx(b)s(e)29 b(a)f(direct)h(sum)g(of)0 2954 y(simple)36 b(ideals,)h(\014nitely)f (man)m(y)-8 b(,)36 b(since)g Fv(R)g Fx(is)f(\014nitely)h(generated,)h (and)d(let)i Fv(I)2859 2969 y Fr(ij)2951 2927 y Fq(\030)2951 2959 y Fx(=)3059 2954 y Fv(I)3102 2969 y Fr(ik)3204 2954 y Fx(for)e(all)h Fv(i;)17 b(j;)g(k)38 b Fx(and)0 3083 y Fv(I)43 3098 y Fr(i)p Fk(1)134 3083 y Fq(6)134 3055 y(\030)135 3087 y Fx(=)239 3083 y Fv(I)282 3098 y Fr(j)t Fk(1)387 3083 y Fx(for)32 b Fv(i)c Fq(6)p Fx(=)f Fv(j)6 b Fx(.)44 b(Let)32 b Fv(B)1065 3098 y Fr(k)1136 3083 y Fx(:=)1266 3008 y Fl(L)1377 3034 y Fr(i)1401 3046 y Fi(k)1377 3112 y Fr(j)t Fk(=1)1520 3083 y Fv(I)1563 3098 y Fr(k)r(j)1639 3083 y Fx(.)0 3202 y(Let)g Fv(I)j Fq(\022)29 b Fv(R)j Fx(b)s(e)g(simple.)45 b(Let)32 b Fv(p)1161 3217 y Fr(k)1231 3202 y Fx(:)c Fv(R)h Fq(\000)-60 b(!)27 b Fv(B)1607 3217 y Fr(k)1682 3202 y Fx(b)s(e)32 b(the)g(pro)5 b(jection)32 b(on)m(to)g Fv(B)2736 3217 y Fr(k)2810 3202 y Fx(w.r.t.)44 b Fv(R)29 b Fx(=)e Fv(B)3361 3217 y Fk(1)3421 3202 y Fq(\010)21 b Fv(:)c(:)g(:)j Fq(\010)h Fv(B)3826 3217 y Fr(n)3873 3202 y Fx(.)0 3319 y(Then)39 b(there)f(is)g(at)g (least)g(one)g Fv(k)i Fx(with)f Fv(p)1527 3334 y Fr(k)1569 3319 y Fx(\()p Fv(I)8 b Fx(\))36 b Fq(6)p Fx(=)g(0.)59 b(Then)39 b Fv(I)2326 3291 y Fq(\030)2326 3323 y Fx(=)2439 3319 y Fv(p)2488 3334 y Fr(k)2531 3319 y Fx(\()p Fv(I)8 b Fx(\))36 b(=)g Fv(J)46 b Fq(\022)36 b Fv(B)3093 3334 y Fr(k)3174 3319 y Fx(is)i(a)f(simple)i(ideal.)0 3435 y(Because)46 b(of)d(8.17)h(w)m(e)h(get)f Fv(I)38 b Fq(\010)30 b Fx(\()1280 3360 y Fl(L)1391 3386 y Fr(m)1391 3464 y(j)t Fk(=)p Fr(r)r Fk(+1)1623 3435 y Fv(I)1666 3450 y Fr(k)r(j)1741 3435 y Fx(\))47 b(=)g Fv(B)2023 3450 y Fr(k)2113 3435 y Fx(=)g Fv(I)2279 3450 y Fr(k)r Fk(1)2387 3435 y Fq(\010)31 b Fv(:)17 b(:)g(:)29 b Fq(\010)i Fv(I)2790 3450 y Fr(k)r(r)2896 3435 y Fq(\010)g Fx(\()3042 3360 y Fl(L)3152 3386 y Fr(m)3152 3464 y(j)t Fk(=)p Fr(r)r Fk(+1)3384 3435 y Fv(I)3427 3450 y Fr(k)r(j)3502 3435 y Fx(\))44 b(using)h(a)0 3554 y(suitable)34 b(n)m(um)m(b)s(ering.)47 b(Hence)35 b Fv(J)1272 3527 y Fq(\030)1273 3558 y Fx(=)1378 3554 y Fv(I)1421 3569 y Fr(k)r Fk(1)1522 3554 y Fq(\010)23 b Fv(:)17 b(:)g(:)22 b Fq(\010)h Fv(I)1902 3569 y Fr(k)r(r)2012 3554 y Fx(and)33 b(th)m(us)i Fv(r)c Fx(=)e(1)k(and)g Fv(I)2950 3527 y Fq(\030)2951 3558 y Fx(=)3056 3554 y Fv(J)3148 3527 y Fq(\030)3149 3558 y Fx(=)3255 3554 y Fv(I)3298 3569 y Fr(k)r Fk(1)3376 3554 y Fx(.)45 b(So)33 b(there)h(is)0 3671 y(a)i(unique)h Fv(k)i Fx(with)e Fv(p)770 3686 y Fr(k)812 3671 y Fx(\()p Fv(I)8 b Fx(\))34 b Fq(6)p Fx(=)f(0.)53 b(In)37 b(particular)f(w)m(e)h(ha)m(v)m(e)g Fv(I)42 b Fq(\022)34 b Fv(B)2435 3686 y Fr(k)2477 3671 y Fx(.)54 b(If)36 b Fv(f)44 b Fx(:)2812 3686 y Fr(R)2870 3671 y Fv(R)34 b Fq(\000)-59 b(!)3129 3686 y Fr(R)3186 3671 y Fv(R)37 b Fx(with)g Fv(f)11 b Fx(\()p Fv(I)d Fx(\))33 b Fq(6)p Fx(=)g(0)0 3787 y(is)41 b(giv)m(en,)i(then)e Fv(f)11 b Fx(\()p Fv(I)d Fx(\))853 3759 y Fq(\030)854 3791 y Fx(=)971 3787 y Fv(I)48 b Fx(is)40 b(simple)i(and)e Fv(f)11 b Fx(\()p Fv(I)d Fx(\))40 b Fq(\022)h Fv(B)2094 3802 y Fr(k)2177 3787 y Fx(for)f(one)g Fv(k)s Fx(.)67 b(So)40 b Fv(f)11 b Fx(\()p Fv(B)2982 3802 y Fr(k)3024 3787 y Fx(\))41 b Fq(\022)g Fv(B)3295 3802 y Fr(k)3378 3787 y Fx(holds)f(for)g(all)0 3903 y Fv(f)e Fq(2)28 b Fx(Hom)384 3918 y Fr(R)441 3903 y Fx(\()p Fv(:R)q(;)17 b(:R)q Fx(\))793 3875 y Fq(\030)794 3907 y Fx(=)898 3903 y Fv(R)q Fx(,)33 b(and)g Fv(B)1297 3918 y Fr(k)1372 3903 y Fx(is)g(a)f(t)m(w)m(o)h(sided)h(ideal.)0 4019 y(Observ)m(e)39 b(that)e Fv(B)669 4034 y Fr(i)697 4019 y Fv(B)771 4034 y Fr(j)843 4019 y Fq(\022)e Fv(B)1029 4034 y Fr(i)1082 4019 y Fq(\\)26 b Fv(B)1248 4034 y Fr(j)1320 4019 y Fx(=)35 b(0.)56 b(F)-8 b(or)36 b(1)f Fq(2)g Fv(R)i Fx(=)e Fv(B)2223 4034 y Fk(1)2287 4019 y Fq(\010)26 b Fv(:)17 b(:)g(:)24 b Fq(\010)i Fv(B)2706 4034 y Fr(n)2790 4019 y Fx(let)37 b(1)e(=)g Fv(e)3175 4034 y Fk(1)3240 4019 y Fx(+)25 b Fv(:)17 b(:)g(:)24 b Fx(+)h Fv(e)3626 4034 y Fr(n)3710 4019 y Fx(with)0 4135 y Fv(e)45 4150 y Fr(i)114 4135 y Fq(2)41 b Fv(B)295 4150 y Fr(i)324 4135 y Fx(.)66 b(F)-8 b(or)40 b Fv(b)h Fq(2)g Fv(B)863 4150 y Fr(i)932 4135 y Fx(w)m(e)g(get)f Fv(e)1298 4150 y Fr(i)1327 4135 y Fv(b)h Fx(=)g(\()p Fv(e)1609 4150 y Fk(1)1676 4135 y Fx(+)27 b Fv(:)17 b(:)g(:)27 b Fx(+)g Fv(e)2069 4150 y Fr(n)2116 4135 y Fx(\)\(0)g(+)g Fv(:)17 b(:)g(:)27 b Fx(+)h Fv(b)f Fx(+)h Fv(:)17 b(:)g(:)27 b Fx(+)g(0\))40 b(=)h Fv(b)g Fx(=)g Fv(be)3563 4150 y Fr(i)3592 4135 y Fx(.)66 b(Th)m(us)0 4252 y Fv(B)74 4267 y Fr(i)138 4252 y Fx(can)36 b(b)s(e)g(considered)h(as)f(ring)g(with)g(unit)g Fv(e)1742 4267 y Fr(i)1770 4252 y Fx(.)53 b(\()p Fv(B)1962 4267 y Fr(i)2026 4252 y Fx(is)36 b(not)g(a)f(subring)i(of)e Fv(R)i Fx(but)f(a)f(quotien)m(t)i(ring)e(of)0 4368 y Fv(R)q Fx(.\))50 b(Since)36 b Fv(B)521 4383 y Fr(i)549 4368 y Fv(B)623 4383 y Fr(j)692 4368 y Fx(=)31 b(0)j(w)m(e)i(ha)m(v)m (e)g(that)f Fv(L)d Fq(\022)g Fv(B)1750 4383 y Fr(i)1813 4368 y Fx(is)j(a)g(\(one)g(sided)h(resp.)51 b(t)m(w)m(o)36 b(sided\))g Fv(B)3270 4383 y Fr(i)3298 4368 y Fx(-ideal)f(of)f Fv(B)3753 4383 y Fr(i)3816 4368 y Fx(i\013)0 4484 y Fv(L)i Fx(is)f(an)g Fv(R)q Fx(-ideal.)52 b(Since)36 b Fv(B)1058 4499 y Fr(i)1118 4484 y Fx(=)c Fv(I)1269 4499 y Fk(1)1333 4484 y Fq(\010)24 b Fv(:)17 b(:)g(:)24 b Fq(\010)g Fv(I)1717 4499 y Fr(n)1799 4484 y Fx(is)36 b(a)f(direct)h(sum)g(of)f(simple)h Fv(R)q Fx(-ideals)g(resp.)52 b Fv(B)3600 4499 y Fr(i)3628 4484 y Fx(-ideals)0 4600 y(and)38 b(since)i Fv(I)483 4615 y Fr(j)557 4573 y Fq(\030)558 4605 y Fx(=)672 4600 y Fv(I)715 4615 y Fr(k)796 4600 y Fx(holds,)g Fv(B)1159 4615 y Fr(i)1226 4600 y Fx(is)f(a)f(simple)h(ring)g(b)m(y)g(Theorem)g (8.7.)61 b(In)38 b(particular)h Fv(B)3361 4615 y Fr(i)3427 4600 y Fx(has)g(no)f(t)m(w)m(o)0 4717 y(sided)d(non)m(trivial)f (ideals,)h(i.e.)47 b(the)34 b(t)m(w)m(o)g(sided)h(ideals)f Fv(B)2118 4732 y Fr(i)2176 4717 y Fq(\022)29 b Fv(R)35 b Fx(are)f(minimal.)47 b(8.12)33 b(implies)i(that)e Fv(R)h Fx(is)0 4833 y(Artinian.)0 4949 y(\(8\))46 b(=)-17 b Fq(\))47 b Fx(\(7\):)71 b(Since)48 b Fv(B)943 4964 y Fr(i)971 4949 y Fv(B)1045 4964 y Fr(j)1133 4949 y Fq(\022)k Fv(B)1336 4964 y Fr(i)1396 4949 y Fq(\\)32 b Fv(B)1568 4964 y Fr(j)1656 4949 y Fx(=)52 b(0)46 b(the)h Fv(B)2135 4964 y Fr(i)2210 4949 y Fx(are)g(simple)h(rings)f(as)f(ab)s(o)m(v)m(e,) 51 b(hence)d Fv(R)53 b Fx(=)0 5065 y Fv(R)74 5080 y Fk(1)139 5065 y Fq(\002)26 b Fv(:)17 b(:)g(:)25 b Fq(\002)h Fv(R)559 5080 y Fr(n)643 5065 y Fx(with)38 b Fv(R)944 5080 y Fr(i)1008 5065 y Fx(=)e Fv(B)1194 5080 y Fr(i)1222 5065 y Fx(,)j(b)s(ecause)g (addition)e(and)h(m)m(ultiplication)g(are)g(p)s(erformed)f(in)h(the)g Fv(B)3872 5080 y Fr(i)0 5182 y Fx(\(comp)s(onen)m(t)m(wise\).)0 5298 y(\(7\))32 b(=)-17 b Fq(\))33 b Fx(\(1\):)43 b(Lemma)33 b(8.12.)0 5414 y(\(7\))j(=)-17 b Fq(\))36 b Fx(\(9\):)50 b(In)36 b(order)g(to)g(ha)m(v)m(e)h(condition)g(\(7\))f(symmetric)i(in) e(the)h(sides,)h(it)e(su\016ces)i(to)e(sho)m(w)h(that)0 5530 y(a)44 b(simple)j(ring)d Fv(R)i Fx(is)f(righ)m(t)g(Artinian.)80 b(But)45 b Fv(R)1896 5503 y Fq(\030)1897 5534 y Fx(=)2022 5530 y Fv(M)2116 5545 y Fr(n)2163 5530 y Fx(\()p Fv(D)s Fx(\))2371 5503 y Fq(\030)2372 5534 y Fx(=)2497 5530 y(Hom)2700 5545 y Fr(D)2764 5530 y Fx(\()p Fv(V)2880 5494 y Fp(\003)2920 5530 y Fv(:;)17 b(V)5 b(:)3080 5494 y Fp(\003)3119 5530 y Fx(\))45 b(is)g(left)g(and)g(righ)m(t)0 5646 y(Artinian.)3436 b Fo(\003)p eop end %%Page: 73 73 TeXDict begin 73 72 bop 1239 -170 a Fu(Simple)26 b(and)f(Semisimple)h (rings)h(and)e(Mo)r(dules)1163 b(73)0 29 y Fx(8.4.)49 b Ft(No)s(etherian)37 b(Mo)s(dules.)0 208 y(De\014nition)45 b(8.23.)g Fx(A)38 b(mo)s(dule)1266 223 y Fr(F)1325 208 y Fv(M)49 b Fx(is)39 b(called)g Fs(No)-5 b(etherian)38 b Fx(\(Emm)m(y)i(No)s(ether)e(1882-1935\),)g(if)g(eac)m(h)0 324 y(nonempt)m(y)c(set)g(of)e(submo)s(dules)i(of)e Fv(M)43 b Fx(has)33 b(a)g(maximal)g(elemen)m(t.)0 502 y Ft(Theorem)38 b(8.24.)k Fs(F)-7 b(or)929 517 y Fr(R)987 502 y Fv(M)45 b Fs(the)35 b(fol)5 b(lowing)34 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)148 642 y Fx(\(1\))42 b Fv(M)j Fs(is)35 b(No)-5 b(etherian.)148 759 y Fx(\(2\))42 b Fs(Each)33 b(asc)-5 b(ending)32 b(chain)g Fv(M)1357 774 y Fr(i)1414 759 y Fq(\022)c Fv(M)1613 774 y Fr(i)p Fk(+1)1731 759 y Fv(;)17 b(i)28 b Fq(2)g Fn(N)34 b Fs(of)f(submo)-5 b(dules)33 b(of)g Fv(M)44 b Fs(b)-5 b(e)g(c)g(omes)33 b(stationary,)h(i.e.)315 875 y(ther)-5 b(e)34 b(is)h(an)f Fv(n)28 b Fq(2)g Fn(N)35 b Fs(with)g Fv(M)1394 890 y Fr(n)1469 875 y Fx(=)28 b Fv(M)1667 890 y Fr(n)p Fk(+)p Fr(i)1828 875 y Fs(for)34 b(al)5 b(l)35 b Fv(i)28 b Fq(2)g Fn(N)p Fs(.)148 991 y Fx(\(3\))42 b Fs(Each)34 b(submo)-5 b(dule)34 b(of)h Fv(M)45 b Fs(is)35 b(\014nitely)g(gener)-5 b(ate)g(d.)0 1170 y(Pr)g(o)g(of.)41 b Fx(\(2\))c(=)-17 b Fq(\))37 b Fx(\(1\):)53 b(Let)37 b Fq(M)g Fx(b)s(e)h(a)f(nonempt)m(y) i(set)f(of)f(submo)s(dules)i(without)f(a)f(maximal)h(elemen)m(t.)0 1286 y(Using)28 b(the)h(axiom)f(of)f(c)m(hoice)i(w)m(e)g(c)m(ho)s(ose)g (for)e(eac)m(h)h Fv(N)39 b Fq(2)28 b(M)f Fx(an)h Fv(N)2488 1250 y Fp(0)2539 1286 y Fq(2)g(M)g Fx(with)g Fv(N)38 b Fn($)28 b Fv(N)3307 1250 y Fp(0)3331 1286 y Fx(.)42 b(F)-8 b(or)27 b Fv(N)38 b Fq(2)28 b(M)0 1402 y Fx(w)m(e)34 b(then)f(ha)m(v)m(e)h(an)e(ascending)i(c)m(hain)g Fv(M)1523 1417 y Fk(1)1590 1402 y Fx(=)28 b Fv(N)5 b(;)17 b(M)1915 1417 y Fr(i)p Fk(+1)2061 1402 y Fx(=)27 b Fv(M)2268 1366 y Fp(0)2258 1427 y Fr(i)2325 1402 y Fx(with)1202 1566 y Fv(M)1296 1581 y Fk(1)1363 1566 y Fn($)i Fv(M)1563 1581 y Fk(2)1630 1566 y Fn($)f Fv(:)17 b(:)g(:)27 b Fn($)i Fv(M)2077 1581 y Fr(i)2133 1566 y Fn($)f Fv(M)2332 1581 y Fr(i)p Fk(+1)2478 1566 y Fn($)h Fv(:)17 b(:)g(:)0 1730 y Fx(This)34 b(is)f(imp)s(ossible)h(b)m(y)g(\(2\).)0 1846 y(\(1\))41 b(=)-17 b Fq(\))40 b Fx(\(3\):)60 b(Let)41 b Fv(M)864 1810 y Fp(0)930 1846 y Fq(\022)i Fv(M)10 b Fx(.)69 b(Then)42 b Fq(f)p Fv(N)10 b Fq(j)p Fv(N)53 b Fq(\022)42 b Fv(M)2033 1810 y Fp(0)2057 1846 y Fv(;)17 b(N)43 b Fx(\014nitely)34 b(generated)q Fq(g)42 b(6)p Fx(=)f Fq(;)g Fx(has)h(a)e(maximal)0 1962 y(elemen)m(t)45 b Fv(N)459 1926 y Fp(0)482 1962 y Fx(.)74 b(If)43 b Fv(N)779 1926 y Fp(0)848 1962 y Fq(6)p Fx(=)i Fv(M)1073 1926 y Fp(0)1097 1962 y Fx(,)g(then)f(there)f(is)h(an)e Fv(m)k Fq(2)f Fv(M)2261 1926 y Fp(0)2314 1962 y Fq(n)29 b Fv(N)2481 1926 y Fp(0)2505 1962 y Fx(.)74 b(So)43 b Fv(N)2840 1926 y Fp(0)2893 1962 y Fx(+)29 b Fv(R)q(m)45 b Fq(\022)h Fv(M)3430 1926 y Fp(0)3496 1962 y Fx(is)e(\014nitely)0 2079 y(generated)30 b(and)g Fv(N)716 2042 y Fp(0)767 2079 y Fn($)e Fv(N)960 2042 y Fp(0)1000 2079 y Fx(+)16 b Fv(R)q(m)29 b Fx(in)g(con)m(tradiction)i(to)e(the)g(maximalit)m(y)i (of)e Fv(N)2967 2042 y Fp(0)2991 2079 y Fx(.)43 b(Hence)30 b Fv(N)3435 2042 y Fp(0)3487 2079 y Fx(=)d Fv(M)3694 2042 y Fp(0)3718 2079 y Fx(,)j(i.e.)0 2195 y Fv(M)104 2159 y Fp(0)161 2195 y Fx(is)j(\014nitely)h(generated.)0 2311 y(\(3\))g(=)-17 b Fq(\))34 b Fx(\(2\):)47 b(Let)34 b Fv(M)821 2326 y Fk(1)892 2311 y Fq(\022)d Fv(M)1094 2326 y Fk(2)1165 2311 y Fq(\022)g Fv(:)17 b(:)g(:)30 b Fq(\022)i Fv(M)1621 2326 y Fr(n)1699 2311 y Fq(\022)f Fv(:)17 b(:)g(:)30 b Fq(\022)h Fv(M)45 b Fx(b)s(e)35 b(an)f(ascending)i(c)m(hain)f(of)f(submo)s(dules)i(of)0 2427 y Fv(M)10 b Fx(.)43 b(Let)30 b Fv(N)38 b Fx(:=)592 2353 y Fl(S)675 2456 y Fr(i)p Fp(2)p Fm(N)815 2427 y Fv(M)909 2442 y Fr(i)938 2427 y Fx(.)k Fv(N)e Fx(is)29 b(a)g(\014nitely)i(generated)f(submo)s(dule)h(of)d Fv(M)10 b Fx(,)31 b(i.e.)43 b Fv(N)38 b Fx(=)28 b Fv(R)q(a)3333 2442 y Fk(1)3388 2427 y Fx(+)15 b Fv(:)i(:)g(:)e Fx(+)g Fv(R)q(a)3826 2442 y Fr(n)3873 2427 y Fx(.)0 2543 y(Then)36 b(there)g(is)f(an)g Fv(M)841 2558 y Fr(r)914 2543 y Fx(with)h Fv(a)1190 2558 y Fk(1)1229 2543 y Fv(;)17 b(:)g(:)g(:)f(;)h(a)1499 2558 y Fr(n)1577 2543 y Fq(2)32 b Fv(M)1769 2558 y Fr(r)1808 2543 y Fx(.)50 b(This)36 b(implies)h Fv(M)2538 2558 y Fr(r)2608 2543 y Fx(=)31 b Fv(N)42 b Fx(=)32 b Fv(M)3037 2558 y Fr(r)r Fk(+)p Fr(i)3189 2543 y Fx(for)i(all)h Fv(i)d Fq(2)g Fn(N)p Fx(,)j(i.e.)0 2660 y(the)e(c)m(hain)g(b)s(ecomes)h (stationary)-8 b(.)2560 b Fo(\003)0 2866 y Ft(Lemma)50 b(8.25.)d Fs(L)-5 b(et)44 b Fx(0)f Fq(\000)-57 b(!)44 b Fv(M)1306 2806 y Fr(f)1267 2866 y Fq(\000)-58 b(!)44 b Fv(N)1604 2806 y Fr(g)1562 2866 y Fq(\000)-57 b(!)44 b Fv(P)57 b Fq(\000)-57 b(!)44 b Fx(0)f Fs(b)-5 b(e)43 b(a)h(short)f(exact)h(se)-5 b(quenc)g(e.)70 b Fv(M)54 b Fs(and)43 b Fv(P)57 b Fs(ar)-5 b(e)0 2982 y(No)g(etherian)31 b(i\013)g Fv(N)42 b Fs(is)31 b(No)-5 b(etherian.)44 b(In)31 b(p)-5 b(articular)31 b(if)g Fv(M)42 b Fs(and)31 b Fv(N)42 b Fs(ar)-5 b(e)31 b(No)-5 b(etherian)32 b(then)f(so)g(is)g Fv(M)26 b Fq(\010)15 b Fv(N)10 b Fs(.)0 3161 y(Pr)-5 b(o)g(of.)41 b Fx(Let)49 b Fv(N)59 b Fx(b)s(e)49 b(No)s(etherian.)93 b(Then)49 b(it)g(is)g(clear)g(that)g Fv(M)59 b Fx(No)s(etherian.)93 b(If)48 b Fq(f)p Fv(L)3314 3176 y Fr(i)3343 3161 y Fq(g)g Fx(is)i(a)e(set)h(of)0 3277 y(submo)s(dules)40 b(of)d Fv(P)51 b Fx(then)38 b Fq(f)p Fv(g)1093 3241 y Fp(\000)p Fk(1)1187 3277 y Fx(\()p Fv(L)1291 3292 y Fr(i)1319 3277 y Fx(\))p Fq(g)g Fx(is)g(a)f(set)i(of)e(submo)s(dules)j(of)d Fv(N)10 b Fx(.)59 b(Let)38 b Fv(g)2964 3241 y Fp(\000)p Fk(1)3058 3277 y Fx(\()p Fv(L)3162 3292 y Fk(0)3202 3277 y Fx(\))f(b)s(e)h(maximal)h(in)0 3393 y(this)33 b(set.)44 b(With)33 b Fv(g)t(g)734 3357 y Fp(\000)p Fk(1)827 3393 y Fx(\()p Fv(L)931 3408 y Fr(i)960 3393 y Fx(\))27 b(=)h Fv(L)1195 3408 y Fr(i)1256 3393 y Fx(w)m(e)34 b(get)e(that)h Fv(L)1840 3408 y Fk(0)1912 3393 y Fx(is)g(maximal)g(in)g Fq(f)p Fv(L)2638 3408 y Fr(i)2666 3393 y Fq(g)p Fx(.)0 3509 y(Let)f Fv(M)43 b Fx(and)32 b Fv(P)46 b Fx(b)s(e)32 b(No)s(etherian.)44 b(Let)32 b Fq(f)p Fv(L)1573 3524 y Fr(i)1601 3509 y Fq(g)g Fx(b)s(e)g(a)g(set)h(of)e(submo)s(dules)j(of) e Fv(N)10 b Fx(.)44 b(Let)32 b Fv(L)3197 3524 y Fk(0)3269 3509 y Fx(b)s(e)g(c)m(hosen)h(suc)m(h)0 3626 y(that)i Fv(g)t Fx(\()p Fv(L)369 3641 y Fk(0)408 3626 y Fx(\))g(is)g(maximal)h (in)g(the)f(set)h Fq(f)p Fv(g)t Fx(\()p Fv(L)1629 3641 y Fr(i)1656 3626 y Fx(\))p Fq(g)p Fx(.)51 b(Let)35 b Fv(L)h Fx(b)s(e)f(c)m(hosen)i(suc)m(h)f(that)f Fv(f)3046 3590 y Fp(\000)p Fk(1)3140 3626 y Fx(\()p Fv(L)p Fx(\))g(is)h(maximal)g (in)0 3742 y(the)h(set)g Fq(f)p Fv(f)437 3706 y Fp(\000)p Fk(1)530 3742 y Fx(\()p Fv(L)634 3757 y Fr(j)671 3742 y Fx(\))p Fq(j)p Fv(L)803 3757 y Fr(j)874 3742 y Fq(2)d(f)p Fv(L)1090 3757 y Fr(i)1118 3742 y Fq(g)f Fx(and)f Fv(g)t Fx(\()p Fv(L)1545 3757 y Fr(j)1582 3742 y Fx(\))h(=)h Fv(g)t Fx(\()p Fv(L)1918 3757 y Fk(0)1957 3742 y Fx(\))p Fq(g)p Fx(.)54 b(W)-8 b(e)37 b(sho)m(w)g(that)f Fv(L)h Fx(is)g(maximal)g(in)f Fq(f)p Fv(L)3598 3757 y Fr(i)3627 3742 y Fq(g)p Fx(.)54 b(Let)0 3858 y Fv(L)66 3822 y Fp(0)123 3858 y Fq(2)33 b(f)p Fv(L)338 3873 y Fr(i)367 3858 y Fq(g)i Fx(with)i Fv(L)c Fq(\022)h Fv(L)954 3822 y Fp(0)977 3858 y Fx(.)53 b(Then)37 b Fv(g)t Fx(\()p Fv(L)1470 3873 y Fk(0)1509 3858 y Fx(\))c(=)g Fv(g)t Fx(\()p Fv(L)p Fx(\))g Fq(\022)g Fv(g)t Fx(\()p Fv(L)2180 3822 y Fp(0)2203 3858 y Fx(\))j(hence)h Fv(g)t Fx(\()p Fv(L)2706 3822 y Fp(0)2729 3858 y Fx(\))c(=)g Fv(g)t Fx(\()p Fv(L)3064 3873 y Fk(0)3103 3858 y Fx(\).)53 b(F)-8 b(urthermore)36 b(w)m(e)0 3974 y(ha)m(v)m(e)e Fv(f)284 3938 y Fp(\000)p Fk(1)378 3974 y Fx(\()p Fv(L)p Fx(\))28 b Fq(\022)g Fv(f)712 3938 y Fp(\000)p Fk(1)806 3974 y Fx(\()p Fv(L)910 3938 y Fp(0)934 3974 y Fx(\))k(hence)i Fv(L)28 b Fx(=)g Fv(L)1539 3938 y Fp(0)1562 3974 y Fx(.)2234 b Fo(\003)0 4153 y Ft(Corollary)31 b(8.26.)757 4168 y Fr(R)815 4153 y Fv(R)e Fs(is)f(No)-5 b(etherian)28 b(as)g(a)g(left)h Fv(R)q Fs(-mo)-5 b(dule)27 b(i\013)h(al)5 b(l)28 b(\014nitely)g(gener)-5 b(ate)g(d)28 b(left)g Fv(R)q Fs(-mo)-5 b(dules)0 4269 y(ar)g(e)35 b(No)-5 b(etherian.)0 4447 y(Pr)g(o)g(of.)41 b Fq(\()-17 b Fx(=:)43 b(trivial.)0 4564 y(=)-17 b Fq(\))p Fx(:)44 b(If)33 b Fv(M)44 b Fx(is)34 b(\014nitely)g(generated)g(then)g (there)f(is)h(a)f(short)g(exact)h(sequence)i(0)28 b Fq(\000)-60 b(!)28 b Fv(K)36 b Fq(\000)-60 b(!)29 b Fv(R)23 b Fq(\010)g Fv(:)17 b(:)g(:)22 b Fq(\010)h Fv(R)0 4680 y Fq(\000)-60 b(!)28 b Fv(M)38 b Fq(\000)-59 b(!)27 b Fx(0.)41 b(Since)28 b Fv(R)f Fx(is)g(No)s(etherian)g Fv(R)11 b Fq(\010)f Fv(:)17 b(:)g(:)10 b Fq(\010)g Fv(R)28 b Fx(No)s(etherian,)g(to)s(o,)f (so)g(that)f Fv(M)37 b Fx(is)27 b(No)s(etherian.)99 b Fo(\003)0 4858 y Ft(Theorem)28 b(8.27.)35 b Fx(\(Hilb)s(ert)25 b(Basis)g(Theorem\))j Fs(If)e Fv(R)i Fs(is)f(left)g(No)-5 b(etherian)26 b(then)h Fv(R)q Fx([)p Fv(x)p Fx(])h Fs(is)e(left)h(No)-5 b(etherian.)0 5036 y(Pr)g(o)g(of.)41 b Fx(Let)32 b Fv(J)37 b Fq(\022)28 b Fv(R)q Fx([)p Fv(x)p Fx(])k(b)s(e)g(an)f(ideal.)44 b(W)-8 b(e)32 b(ha)m(v)m(e)h(to)e(sho)m(w)h(that)g Fv(J)40 b Fx(\014nitely)33 b(generated.)44 b(Let)32 b Fv(J)3512 5051 y Fk(0)3579 5036 y Fx(:=)27 b Fq(f)p Fv(r)k Fq(2)0 5153 y Fv(R)q Fq(j9)p Fv(p)p Fx(\()p Fv(x)p Fx(\))d Fq(2)g Fv(J)42 b Fx(with)33 b(highest)h(co)s(e\016cien)m(t)g Fv(r)s Fq(g)p Fx(.)43 b(\(The)33 b(highest)g(co)s(e\016cien)m(t)h(of)e (the)g(zero)h(p)s(olynomial)f(is)h(0)0 5269 y(b)m(y)h(de\014nition.\)) 47 b Fv(J)702 5284 y Fk(0)770 5269 y Fq(\022)29 b Fv(R)34 b Fx(is)g(an)f(ideal,)h(hence)h Fv(J)1806 5284 y Fk(0)1874 5269 y Fx(=)29 b Fq(h)p Fv(r)2062 5284 y Fk(1)2101 5269 y Fv(;)17 b(:)g(:)g(:)f(;)h(r)2364 5284 y Fr(n)2411 5269 y Fq(i)p Fx(.)45 b(F)-8 b(or)32 b(the)i Fv(r)2910 5284 y Fr(i)2971 5269 y Fx(c)m(ho)s(ose)g Fv(p)3330 5284 y Fr(i)3359 5269 y Fx(\()p Fv(x)p Fx(\))29 b Fq(2)g Fv(J)42 b 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y Fv(R)19 b Fx(+)e Fv(R)q(x)h Fx(+)g Fv(:)f(:)g(:)g Fx(+)g Fv(R)q(x)783 138 y Fr(m)p Fp(\000)p Fk(1)941 174 y Fx(.)42 b(Both)31 b Fv(R)q Fx(-mo)s(dules)g (are)f(\014nitely)i(generated)f(hence)p 2935 119 V 31 w Fv(g)g Fx(=)3117 100 y Fl(P)3222 126 y Fr(k)3222 203 y(i)p Fk(=1)3357 174 y Fv(\026)3416 189 y Fr(i)3444 174 y Fv(q)3487 189 y Fr(i)3515 174 y Fx(\()p Fv(X)8 b Fx(\))30 b(with)0 291 y Fq(h)p Fv(q)82 306 y Fk(1)121 291 y Fx(\()p Fv(x)p Fx(\))p Fv(;)17 b(:)g(:)g(:)g(;)g(q)515 306 y Fr(k)557 291 y Fx(\()p Fv(x)p Fx(\))p Fq(i)34 b Fx(=)g Fv(J)g Fq(\\)25 b Fx(\()p Fv(R)h Fx(+)e Fv(R)q(x)i Fx(+)e Fv(:)17 b(:)g(:)24 b Fx(+)h Fv(R)q(x)1914 254 y Fr(m)p Fp(\000)p Fk(1)2071 291 y Fx(\).)55 b(Th)m(us)38 b Fq(f)p Fv(p)2541 306 y Fk(1)2580 291 y Fx(\()p Fv(x)p Fx(\))p Fv(;)17 b(:)g(:)g(:)f(;)h(p)2979 306 y Fr(n)3025 291 y Fx(\()p Fv(x)p Fx(\))p Fv(;)g(q)3243 306 y Fk(1)3283 291 y Fx(\()p Fv(x)p Fx(\))p Fv(;)g(:)g(:)g(:)f(;)h(q)3676 306 y Fr(k)3719 291 y Fx(\()p Fv(x)p Fx(\))p Fq(g)0 407 y 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Fr(n)2590 827 y Fs(\).)45 b(Then)34 b Fv(S)40 b Fs(is)35 b(No)-5 b(etherian.)0 1011 y(Pr)g(o)g(of.)41 b Fx(1.)i(By)33 b(induction)h(w)m(e)g(ha)m(v)m(e)g Fv(R)q Fx([)p Fv(x)1530 1026 y Fk(1)1570 1011 y Fv(;)17 b(:)g(:)g(:)e(;)i(x) 1843 1026 y Fr(n)1890 1011 y Fx(])33 b(No)s(etherian.)0 1127 y(2.)40 b(There)25 b(is)f(an)g(epimorphism)h Fv(R)q Fx([)p Fv(x)1333 1142 y Fk(1)1373 1127 y Fv(;)17 b(:)g(:)g(:)f(;)h(x) 1647 1142 y Fr(n)1694 1127 y Fx(])28 b Fq(\000)-60 b(!)28 b Fv(S)6 b Fx(.)40 b(Th)m(us)25 b Fv(S)k Fx(is)24 b(a)g(No)s(etherian)g Fv(R)q Fx([)p Fv(x)3168 1142 y Fk(1)3208 1127 y Fv(;)17 b(:)g(:)g(:)f(;)h(x)3482 1142 y Fr(n)3529 1127 y Fx(]-mo)s(dule)0 1243 y(hence)34 b(it)f(is)g(also)f(a)h(No)s(etherian)g Fv(S)6 b Fx(-mo)s(dule.)2138 b Fo(\003)0 1427 y Ft(Prop)s(osition)47 b(8.29.)e Fs(L)-5 b(et)43 b Fv(R)g Fs(b)-5 b(e)42 b(c)-5 b(ommutative)41 b(or)h Fv(M)53 b Fs(b)-5 b(e)42 b(No)-5 b(etherian.)66 b(L)-5 b(et)42 b Fv(M)53 b Fs(b)-5 b(e)42 b(\014nitely)g(gener-)0 1543 y(ate)-5 b(d.)66 b(L)-5 b(et)43 b Fv(f)52 b Fx(:)41 b Fv(N)52 b Fq(\000)-57 b(!)41 b Fv(M)53 b Fs(b)-5 b(e)41 b(an)h(epimorphism)f(wher)-5 b(e)41 b Fv(N)52 b Fq(\022)41 b Fv(M)53 b Fs(is)42 b(a)g(submo)-5 b(dule.)66 b(Then)42 b Fv(f)52 b Fs(is)42 b(an)0 1660 y(isomorphism.)0 1843 y(Pr)-5 b(o)g(of.)41 b Fx(1.)66 b(Let)40 b Fv(M)51 b Fx(b)s(e)40 b(No)s(etherian.)66 b(W)-8 b(e)41 b(construct)g(an)f(ascending)h(c)m(hain)g Fv(K)3026 1858 y Fk(0)3106 1843 y Fq(\022)g Fv(K)3307 1858 y Fk(1)3387 1843 y Fq(\022)g Fv(K)3588 1858 y Fk(2)3668 1843 y Fq(\022)g Fv(:)17 b(:)g(:)0 1959 y Fx(b)m(y)40 b Fv(K)225 1974 y Fk(0)303 1959 y Fx(:=)e(Ke\()p Fv(f)11 b Fx(\))38 b(=)h Fv(f)910 1923 y Fp(\000)p Fk(1)1004 1959 y Fx(\(0\),)h Fv(K)1279 1974 y Fr(i)1345 1959 y Fx(:=)f Fv(f)1546 1923 y Fp(\000)p Fk(1)1640 1959 y Fx(\()p Fv(K)1761 1974 y Fr(i)p Fp(\000)p Fk(1)1879 1959 y Fx(\).)62 b(W)-8 b(e)40 b(ha)m(v)m(e)g Fv(K)2495 1974 y Fk(0)2573 1959 y Fx(=)e Fv(f)2746 1923 y Fp(\000)p Fk(1)2840 1959 y Fx(\(0\))g Fq(\022)h Fv(f)3178 1923 y Fp(\000)p Fk(1)3272 1959 y Fx(\()p Fv(K)3393 1974 y Fk(0)3433 1959 y Fx(\))f(=)g Fv(K)3706 1974 y Fk(1)3746 1959 y Fx(.)62 b(If)0 2076 y Fv(K)83 2091 y Fr(i)p Fp(\000)p Fk(2)229 2076 y Fq(\022)28 b Fv(K)417 2091 y Fr(i)p Fp(\000)p Fk(1)565 2076 y Fx(then)i(w)m(e)g (ha)m(v)m(e)h Fv(K)1229 2091 y Fr(i)p Fp(\000)p Fk(1)1375 2076 y Fx(=)c Fv(f)1537 2039 y Fp(\000)p Fk(1)1631 2076 y Fx(\()p Fv(K)1752 2091 y Fr(i)p Fp(\000)p Fk(2)1871 2076 y Fx(\))g Fq(\022)h Fv(f)2100 2039 y Fp(\000)p Fk(1)2194 2076 y Fx(\()p Fv(K)2315 2091 y Fr(i)p Fp(\000)p Fk(1)2434 2076 y Fx(\))f(=)h Fv(K)2686 2091 y Fr(i)2714 2076 y Fx(.)42 b(Since)31 b Fv(M)40 b Fx(is)30 b(No)s(etherian)g(the)0 2192 y(c)m(hain)35 b(b)s(ecomes)h(stationary)e Fv(K)1197 2207 y Fr(n)1274 2192 y Fx(=)c Fv(K)1463 2207 y Fr(n)p Fk(+1)1631 2192 y Fx(=)g Fv(:)17 b(:)g(:)o Fx(.)48 b(Let)35 b Fv(x)2158 2207 y Fk(0)2228 2192 y Fq(2)c Fv(K)2408 2207 y Fk(0)2447 2192 y Fx(.)48 b(W)-8 b(e)35 b(w)m(an)m(t)g(to)e(sho)m (w)j Fv(x)3351 2207 y Fk(0)3421 2192 y Fx(=)30 b(0.)48 b(There)0 2308 y(is)40 b Fv(x)160 2323 y Fk(1)240 2308 y Fq(2)h Fv(K)430 2323 y Fk(1)509 2308 y Fx(with)g Fv(f)11 b Fx(\()p Fv(x)891 2323 y Fk(1)930 2308 y Fx(\))40 b(=)g Fv(x)1179 2323 y Fk(0)1219 2308 y Fx(,)i(since)f Fv(f)51 b Fx(is)40 b(an)g(epimorphism.)67 b(Similarly)41 b(there)g(are)f Fv(x)3425 2323 y Fk(0)3465 2308 y Fv(;)17 b(x)3564 2323 y Fk(1)3603 2308 y Fv(;)g(x)3702 2323 y Fk(2)3742 2308 y Fv(;)g(:)g(:)g(:)0 2424 y Fx(with)38 b Fv(f)11 b Fx(\()p Fv(x)379 2439 y Fr(i)407 2424 y Fx(\))35 b(=)g Fv(x)646 2439 y Fr(i)p Fp(\000)p Fk(1)802 2424 y Fx(and)i Fv(f)1055 2388 y Fr(n)p Fk(+1)1192 2424 y Fx(\()p Fv(x)1285 2439 y Fr(n)p Fk(+1)1422 2424 y Fx(\))e(=)g Fv(f)1665 2388 y Fr(n)1712 2424 y Fx(\()p Fv(x)1805 2439 y Fr(n)1852 2424 y Fx(\))g(=)g Fv(:)17 b(:)g(:)35 b Fx(=)g Fv(f)11 b Fx(\()p Fv(x)2449 2439 y Fk(1)2489 2424 y Fx(\))35 b(=)g Fv(x)2728 2439 y Fk(0)2768 2424 y Fx(.)56 b(Since)38 b(the)g(c)m(hain)f(b)s(ecomes)0 2541 y(stationary)29 b(w)m(e)h(get)e Fv(x)813 2556 y Fr(n)p Fk(+1)978 2541 y Fq(2)h Fv(K)1156 2556 y Fr(n)1202 2541 y Fx(,)h(whic)m(h)g(implies)g Fv(f)11 b Fx(\()p Fv(x)2013 2556 y Fr(n)p Fk(+1)2150 2541 y Fx(\))28 b Fq(2)g Fv(K)2393 2556 y Fr(n)p Fp(\000)p Fk(1)2559 2541 y Fx(and)h(th)m(us)g Fv(f)3014 2504 y Fr(n)3061 2541 y Fx(\()p Fv(x)3154 2556 y Fr(n)p Fk(+1)3291 2541 y Fx(\))f Fq(2)g Fv(K)3534 2556 y Fk(0)3573 2541 y Fx(.)43 b(Hence)0 2657 y Fv(x)55 2672 y Fk(0)123 2657 y Fx(=)27 b Fv(f)285 2621 y Fr(n)p Fk(+1)422 2657 y Fx(\()p Fv(x)515 2672 y Fr(n)p Fk(+1)652 2657 y Fx(\))h(=)g(0.)43 b(This)34 b(pro)m(v)m(es)g(that)e Fv(f)44 b Fx(is)33 b(a)f(monomorphism.)0 2773 y(2.)43 b(Let)32 b Fv(R)i Fx(comm)m(utativ)m(e.)45 b(Let)32 b Fv(M)39 b Fx(=)27 b Fv(R)q(y)1554 2788 y Fk(1)1615 2773 y Fx(+)21 b Fv(:)c(:)g(:)k Fx(+)g Fv(R)q(y)2068 2788 y Fr(n)2115 2773 y Fx(.)43 b(Let)32 b Fv(x)2414 2788 y Fr(i)2471 2773 y Fq(2)c Fv(N)2643 2788 y Fr(i)2703 2773 y Fx(with)33 b Fv(f)11 b Fx(\()p Fv(x)3077 2788 y Fr(i)3105 2773 y Fx(\))28 b(=)f Fv(y)3322 2788 y Fr(i)3350 2773 y Fx(.)43 b(Let)33 b Fv(x)3650 2788 y Fk(0)3717 2773 y Fq(2)28 b Fv(N)0 2889 y Fx(with)37 b Fv(f)11 b Fx(\()p Fv(x)378 2904 y Fk(0)417 2889 y Fx(\))33 b(=)h(0.)53 b(Then)37 b(there)g(are)f(co)s(e\016cien)m(ts)i Fv(r)1945 2904 y Fr(ij)2039 2889 y Fq(2)c Fv(R)j Fx(with)f Fv(x)2530 2904 y Fr(i)2592 2889 y Fx(=)2702 2815 y Fl(P)2807 2841 y Fr(n)2807 2918 y(j)t Fk(=1)2950 2889 y Fv(r)2994 2904 y Fr(ij)3055 2889 y Fv(y)3103 2904 y Fr(j)3139 2889 y Fv(;)17 b(i)33 b Fx(=)g(0)p Fv(;)17 b(:)g(:)g(:)f(;)h(n)p Fx(.)54 b(W)-8 b(e)0 3014 y(consider)43 b Fv(R)464 2978 y Fp(0)529 3014 y Fx(:=)g Fn(Z)p Fx([)p Fv(r)812 3029 y Fr(ij)873 3014 y Fx(])f Fq(\022)h Fv(R)q Fx(,)h(the)d(subring)h(of)f Fv(R)h Fx(generated)g(b)m(y)g(the)g Fv(r)2794 3029 y Fr(ij)2854 3014 y Fx(.)70 b(Since)42 b Fn(Z)g Fx(is)g(No)s(etherian)0 3130 y(and)32 b Fv(R)264 3094 y Fp(0)320 3130 y Fx(is)g(\014nitely)i (generated)f(as)f(a)g Fn(Z)p Fx(-algebra)g Fv(R)1904 3094 y Fp(0)1959 3130 y Fx(is)h(No)s(etherian.)43 b(Let)33 b Fv(M)2877 3094 y Fp(0)2928 3130 y Fx(:=)3059 3056 y Fl(P)3164 3082 y Fr(n)3164 3160 y(i)p Fk(=1)3299 3130 y Fv(R)3374 3094 y Fp(0)3397 3130 y Fv(y)3445 3145 y Fr(i)3501 3130 y Fq(\022)28 b Fv(M)43 b Fx(and)0 3247 y Fv(N)88 3211 y Fp(0)148 3247 y Fx(=)260 3172 y Fl(P)365 3198 y Fr(n)365 3276 y(i)p Fk(=0)500 3247 y Fv(R)575 3211 y Fp(0)598 3247 y Fv(x)653 3262 y Fr(i)718 3247 y Fq(\022)37 b Fv(N)10 b Fx(.)59 b(Then)38 b Fv(N)1353 3211 y Fp(0)1413 3247 y Fq(\022)f Fv(M)1631 3211 y Fp(0)1692 3247 y Fx(is)h(an)g Fv(R)2011 3211 y Fp(0)2034 3247 y Fx(-submo)s(dule,)i Fv(M)2695 3211 y Fp(0)2756 3247 y Fx(as)e(an)g Fv(R)3097 3211 y Fp(0)3120 3247 y Fx(-mo)s(dule)g(is)g (\014nitely)0 3363 y(generated,)43 b(hence)e(No)s(etherian,)h(and)e (the)g Fv(f)11 b Fx(\()p Fv(x)1823 3378 y Fr(i)1852 3363 y Fx(\))40 b(=)g Fv(y)2094 3378 y Fr(i)2121 3363 y Fv(;)17 b(f)11 b Fx(\()p Fv(x)2317 3378 y Fk(0)2357 3363 y Fx(\))40 b(=)g(0)f(generate)i(a)e(homomorphism)i(of)0 3479 y Fv(R)75 3443 y Fp(0)98 3479 y Fx(-mo)s(dules)35 b Fv(f)574 3443 y Fp(0)627 3479 y Fx(:)30 b Fv(N)772 3443 y Fp(0)825 3479 y Fq(\000)-59 b(!)29 b Fv(M)1076 3443 y Fp(0)1100 3479 y Fx(.)48 b(Since)35 b Fv(f)1490 3443 y Fp(0)1547 3479 y Fx(is)f(surjectiv)m(e)i Fv(f)2143 3443 y Fp(0)2200 3479 y Fx(is)e(injectiv)m(e)i(and)e(th)m(us)h Fv(x)3150 3494 y Fk(0)3220 3479 y Fx(=)30 b(0)j(so)h(that)g Fv(f)44 b Fx(is)0 3595 y(injectiv)m(e.)3443 b Fo(\003)0 3779 y Ft(Problem)46 b(8.1.)f Fx(Where)c(do)s(es)f(the)g(comm)m(utativit)m (y)i(of)d Fv(R)i Fx(en)m(ter)f(the)g(second)h(part)e(of)h(the)g(pro)s (of)e(of)0 3895 y(Prop)s(osition)33 b(8.29?)0 4079 y Ft(Corollary)42 b(8.30.)h Fs(L)-5 b(et)37 b Fv(R)h Fs(b)-5 b(e)36 b(c)-5 b(ommutative)37 b(or)1879 4094 y Fr(R)1936 4079 y Fv(M)48 b Fs(b)-5 b(e)36 b(No)-5 b(etherian.)51 b(L)-5 b(et)37 b Fv(M)42 b Fx(=)31 b Fv(R)q(y)3281 4094 y Fk(1)3344 4079 y Fx(+)23 b Fv(:)17 b(:)g(:)23 b Fx(+)h Fv(R)q(y)3804 4094 y Fr(m)3870 4079 y Fs(.)0 4195 y(L)-5 b(et)37 b Fv(N)j Fq(\022)31 b Fv(M)47 b Fs(b)-5 b(e)36 b(a)g(fr)-5 b(e)g(e)36 b(submo)-5 b(dule)35 b(with)h(the)g(fr)-5 b(e)g(e)36 b(gener)-5 b(ating)36 b(elements)f Fv(x)2894 4210 y Fk(1)2934 4195 y Fv(;)17 b(:)g(:)g(:)f(;)h(x)3208 4210 y Fr(n)3255 4195 y Fs(.)48 b(Then)36 b Fv(n)30 b Fq(\024)h Fv(m)p Fs(.)0 4311 y(If)j Fv(n)28 b Fx(=)g Fv(m)35 b Fs(then)g Fv(M)45 b Fs(is)35 b(fr)-5 b(e)g(e)34 b(over)h Fv(y)1322 4326 y Fk(1)1361 4311 y Fv(;)17 b(:)g(:)g(:)f(;)h(y) 1628 4326 y Fr(m)1694 4311 y Fs(.)0 4495 y(Pr)-5 b(o)g(of.)41 b Fx(Since)47 b Fv(N)57 b Fx(is)46 b(free)g(there)h(is)f(a)g (homomorphism)h Fv(f)61 b Fx(:)50 b Fv(N)61 b Fq(\000)-59 b(!)50 b Fv(M)56 b Fx(with)47 b Fv(f)11 b Fx(\()p Fv(x)3214 4510 y Fr(i)3242 4495 y Fx(\))50 b(=)h Fv(y)3505 4510 y Fr(i)3578 4495 y Fx(for)46 b Fv(i)k Fx(=)0 4611 y(1)p Fv(;)17 b(:)g(:)g(:)f(;)h Fx(min\()p Fv(m;)g(n)p Fx(\))39 b(and)f Fv(f)11 b Fx(\()p Fv(x)1079 4626 y Fr(i)1107 4611 y Fx(\))38 b(=)g(0)g(else.)63 b(If)39 b Fv(n)f Fq(\025)g Fv(m)h Fx(then)g Fv(f)49 b Fx(is)40 b(surjectiv)m(e,)i(hence)e (bijectiv)m(e.)63 b(Th)m(us)0 4727 y(w)m(e)43 b(ha)m(v)m(e)h Fv(n)h Fq(\024)g Fv(m)p Fx(.)73 b(If)42 b Fv(n)j Fx(=)f Fv(m)f Fx(then)g Fv(f)53 b Fx(is)42 b(bijectiv)m(e)j(and)d Fv(M)53 b Fx(free)43 b(with)g(the)g(generating)g(elemen)m(ts)0 4844 y Fv(y)48 4859 y Fk(1)87 4844 y Fv(;)17 b(:)g(:)g(:)f(;)h(y)354 4859 y Fr(n)400 4844 y Fx(.)3396 b Fo(\003)0 5027 y Ft(Corollary)44 b(8.31.)g Fs(L)-5 b(et)40 b Fv(R)f Fs(b)-5 b(e)39 b(c)-5 b(ommutative)38 b(or)h(No)-5 b(etherian.)56 b(L)-5 b(et)39 b Fv(M)50 b Fs(b)-5 b(e)38 b(fr)-5 b(e)g(e)39 b(over)f Fv(x)3346 5042 y Fk(1)3386 5027 y Fv(;)17 b(:)g(:)g(:)f(;)h(x)3660 5042 y Fr(n)3746 5027 y Fs(and)0 5144 y(fr)-5 b(e)g(e)35 b(over)f Fv(y)449 5159 y Fk(1)488 5144 y Fv(;)17 b(:)g(:)g(:)f(;)h(y) 755 5159 y Fr(m)821 5144 y Fs(.)44 b(Then)34 b(we)h(have)f Fv(m)28 b Fx(=)g Fv(n)p Fs(.)0 5327 y(Pr)-5 b(o)g(of.)41 b Fx(If)33 b Fv(R)g Fx(is)g(No)s(etherian)g(then)g Fv(M)43 b Fx(is)33 b(also)f(No)s(etherian.)44 b(Th)m(us)34 b(the)f(claim)g (follo)m(ws)g(from)f(8.30.)98 b Fo(\003)0 5511 y Ft(De\014nition)46 b(8.32.)f Fx(Let)39 b Fv(R)h Fx(b)s(e)f(comm)m(utativ)m(e)j(or)c(No)s (etherian.)64 b(The)40 b Fs(r)-5 b(ank)38 b Fx(of)h(a)g(\014nitely)h (generated)0 5627 y(free)33 b(mo)s(dule)531 5642 y Fr(R)589 5627 y Fv(M)43 b Fx(is)33 b(the)g(n)m(um)m(b)s(er)h(of)e(free)h (generating)g(elemen)m(ts)i(uniquely)g(determined)f(b)m(y)g(8.31.)p eop end %%Page: 75 75 TeXDict begin 75 74 bop 1239 -170 a Fu(Simple)26 b(and)f(Semisimple)h (rings)h(and)e(Mo)r(dules)1163 b(75)0 29 y Ft(Example)37 b(8.33.)j Fx(The)32 b(endomorphism)g(ring)f(of)g(a)f(coun)m(tably)i (in\014nite)g(dimensional)h(v)m(ector)e(space)h(is)0 146 y(neither)i(left)e(nor)h(righ)m(t)f(No)s(etherian.)0 320 y Fs(Pr)-5 b(o)g(of.)41 b Fx(F)-8 b(rom)43 b Fv(ap)30 b Fx(+)f Fv(bq)50 b Fx(=)c(1)p Fv(;)17 b(pa)46 b Fx(=)g(1)p Fv(;)17 b(q)t(b)45 b Fx(=)h(1)p Fv(;)17 b(pb)46 b Fx(=)g(0)p Fv(;)17 b(q)t(a)45 b Fx(=)h(0)d(w)m(e)i(get)e(\(as)g(in)h(the)g (exercise)h(1.4\))0 451 y Fr(R)58 436 y Fv(R)28 b Fx(=)264 451 y Fr(R)322 436 y Fv(R)q(p)22 b Fq(\010)568 451 y Fr(R)625 436 y Fv(R)q(q)37 b Fx(free)c(and)f Fv(R)1230 451 y Fr(R)1316 436 y Fx(=)27 b Fv(aR)1544 451 y Fr(R)1625 436 y Fq(\010)22 b Fv(bR)1839 451 y Fr(R)1930 436 y Fx(free.)1712 b Fo(\003)0 611 y Ft(De\014nition)39 b(8.34.)k Fx(An)34 b(elemen)m(t)h Fv(r)d Fq(2)e Fv(R)35 b Fx(in)f(a)f(ring)h Fv(R)h Fx(is)f(called)g(a)g Fs(left)i(unit)e Fx(\()p Fs(right)h(unit)p Fx(\),)g(if)e Fv(r)s(R)e Fx(=)e Fv(R)0 727 y Fx(\()p Fv(R)q(r)h Fx(=)e Fv(R)q Fx(\).)43 b Fv(r)31 b Fq(2)d Fv(R)33 b Fx(is)g(called)h(a)e Fs(unit)p Fx(,)h(if)g Fv(R)q(r)d Fx(=)e Fv(R)g Fx(=)g Fv(r)s(R)q Fx(.)0 901 y Ft(Lemma)44 b(8.35.)f Fs(If)38 b Fv(r)f Fq(2)e Fv(R)k Fs(is)f(a)h(unit,)g(then)f(ther)-5 b(e)39 b(is)f(a)g(unique)g Fv(s)d Fq(2)f Fv(R)40 b Fs(with)e Fv(sr)f Fx(=)d(1)p Fs(.)55 b(F)-7 b(urthermor)i(e)0 1017 y(we)34 b(have)h Fv(r)s(s)27 b Fx(=)g(1)35 b Fs(and)f Fv(s)h Fs(is)g(a)g(unit.)0 1192 y(Pr)-5 b(o)g(of.)41 b Fx(Let)i Fv(sr)j Fx(=)e Fv(s)785 1156 y Fp(0)808 1192 y Fv(r)j Fx(=)d(1)e(and)g(let)h Fv(r)s(t)h Fx(=)g(1.)72 b(Then)43 b Fv(s)h Fx(=)g Fv(s)p Fx(1)g(=)g Fv(sr)s(t)g Fx(=)g(1)p Fv(t)g Fx(=)g Fv(t)e Fx(and)h(analogously)0 1308 y Fv(s)46 1272 y Fp(0)97 1308 y Fx(=)27 b Fv(t)p Fx(.)3561 b Fo(\003)0 1482 y Ft(Corollary)36 b(8.36.)j Fs(In)31 b(e)-5 b(ach)32 b(left)g(No)-5 b(etherian)31 b(ring)h Fv(R)h Fs(e)-5 b(ach)31 b(right)h(unit)g Fv(x)c Fq(2)g Fv(R)33 b Fs(\(i.e.)43 b Fv(R)q(x)29 b Fx(=)e Fv(R)q Fs(\))32 b(is)g(also)0 1599 y(a)j(left)g(unit)g(and)f(c) -5 b(onversely.)0 1773 y(Pr)g(o)g(of.)41 b Fx(Let)34 b Fv(R)q(x)d Fx(=)f Fv(R)q Fx(.)48 b(Then)35 b Fq(\001)p Fv(x)30 b Fx(:)g Fv(R)i Fq(\000)-60 b(!)30 b Fv(R)35 b Fx(is)f(an)g(epimorphism,)j(hence)e(an)f(isomorphism.)49 b(So)34 b(there)0 1889 y(is)42 b(an)f(in)m(v)m(erse)j(isomorphism)f Fv(g)i Fx(:)e Fv(R)h Fq(\000)-60 b(!)42 b Fv(R)h Fx(with)f Fv(g)j Fq(2)e Fx(Hom)2354 1904 y Fr(R)2411 1889 y Fx(\()p Fv(:R)q(;)17 b(:R)q Fx(\))2778 1861 y Fq(\030)2779 1893 y Fx(=)2898 1889 y Fv(R)q Fx(,)43 b(hence)g Fv(g)j Fx(=)c Fq(\001)p Fv(y)t Fx(.)69 b(This)0 2005 y(implies)34 b(1)22 b Fq(\001)g Fv(x)h Fq(\001)f Fv(y)31 b Fx(=)d(1)33 b(and)g(1)22 b Fq(\001)g Fv(y)j Fq(\001)d Fv(x)29 b Fx(=)f(1,)33 b(i.e.)44 b Fv(x)1799 1969 y Fp(\000)p Fk(1)1922 2005 y Fx(=)28 b Fv(y)36 b Fx(and)d Fv(x)g Fx(is)h(a)e(unit.)45 b(If)32 b Fv(xR)e Fx(=)e Fv(R)34 b Fx(then)f(there)h(is)f(a)0 2122 y Fv(y)e Fq(2)d Fv(R)33 b Fx(with)g Fv(xy)e Fx(=)d(1.)43 b(So)32 b Fv(y)j Fx(is)d(a)g(righ)m(t)g(unit)h(hence)g Fv(y)j Fx(is)c(a)g(unit.)44 b(By)32 b(8.36)g Fv(x)g Fx(is)h(the)f (unique)i(in)m(v)m(erse)g(of)0 2238 y Fv(y)t Fx(,)e(hence)i Fv(x)f Fx(is)g(a)f(unit.)2974 b Fo(\003)p eop end %%Page: 76 76 TeXDict begin 76 75 bop 0 -170 a Fu(76)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)1411 29 y Fx(9.)48 b Fw(Radical)38 b(and)g(Socle)0 204 y Ft(De\014nition)g(9.1.)190 b Fx(\(1\))42 b Fv(N)c Fq(\022)28 b Fv(M)43 b Fx(is)33 b(called)h Fs(lar)-5 b(ge)32 b Fx(\()p Fs(essential)p Fx(\))f(i\013)1247 361 y Fq(8)p Fv(U)39 b Fq(\022)28 b Fv(M)38 b Fx(:)28 b Fv(N)33 b Fq(\\)22 b Fv(U)39 b Fx(=)27 b(0)h(=)-17 b Fq(\))27 b Fv(U)39 b Fx(=)27 b(0)p Fv(:)148 518 y Fx(\(2\))42 b Fv(N)c Fq(\022)28 b Fv(M)43 b Fx(is)33 b(called)h Fs(smal)5 b(l)31 b Fx(\()p Fs(sup)-5 b(er\015uous)p Fx(\))33 b(i\013)1189 675 y Fq(8)p Fv(U)39 b Fq(\022)28 b Fv(M)38 b Fx(:)28 b Fv(N)33 b Fx(+)22 b Fv(U)38 b Fx(=)28 b Fv(M)38 b Fx(=)-17 b Fq(\))28 b Fv(U)38 b Fx(=)28 b Fv(M)5 b(:)0 835 y Ft(Lemma)39 b(9.2.)j Fs(L)-5 b(et)35 b Fv(N)j Fq(\022)29 b Fv(M)38 b Fq(\022)28 b Fv(P)14 b Fs(,)34 b Fv(U)39 b Fq(\022)28 b Fv(P)48 b Fs(b)-5 b(e)35 b(submo)-5 b(dules.)44 b(Then)34 b(the)h Fx(mo)s(dular)d(la)m(w)k Fs(holds:)1297 992 y Fv(N)d Fx(+)22 b(\()p Fv(U)32 b Fq(\\)23 b Fv(M)10 b Fx(\))28 b(=)g(\()p Fv(N)k Fx(+)22 b Fv(U)10 b Fx(\))23 b Fq(\\)g Fv(M)5 b(:)0 1153 y Fs(Pr)-5 b(o)g(of.)41 b Fq(\022)p Fx(:)j(F)-8 b(rom)31 b Fv(n)21 b Fx(+)g Fv(u)27 b Fq(2)h Fv(N)j Fx(+)21 b Fv(U)42 b Fx(with)33 b Fv(n)28 b Fq(2)g Fv(N)42 b Fx(and)32 b Fv(u)27 b Fq(2)h Fv(U)k Fq(\\)21 b Fv(M)38 b Fq(\022)29 b Fv(M)42 b Fx(it)32 b(follo)m(ws)h(that)e Fv(n)21 b Fx(+)g Fv(u)27 b Fq(2)h Fv(M)0 1269 y Fx(and)33 b(hence)h Fv(n)22 b Fx(+)g Fv(u)27 b Fq(2)h Fx(\()p Fv(N)33 b Fx(+)22 b Fv(U)10 b Fx(\))23 b Fq(\\)f Fv(M)10 b Fx(.)0 1386 y Fq(\023)p Fx(:)68 b(F)-8 b(rom)45 b Fv(n)30 b Fx(+)h Fv(u)48 b Fx(=)g Fv(m)h Fq(2)f Fx(\()p Fv(N)41 b Fx(+)31 b Fv(U)10 b Fx(\))31 b Fq(\\)g Fv(M)55 b Fx(it)45 b(follo)m(ws)g(that)g Fv(u)j Fx(=)g Fv(m)31 b Fq(\000)g Fv(n)48 b Fq(2)h Fv(M)41 b Fq(\\)31 b Fv(U)56 b Fx(and)45 b(hence)0 1502 y Fv(n)22 b Fx(+)g Fv(u)28 b Fq(2)g Fv(N)k Fx(+)22 b(\()p Fv(U)33 b Fq(\\)23 b Fv(M)10 b Fx(\).)2864 b Fo(\003)0 1663 y Ft(Lemma)39 b(9.3.)190 b Fx(\(1\))42 b Fs(L)-5 b(et)34 b Fv(N)k Fq(\022)28 b Fv(N)1404 1627 y Fp(0)1455 1663 y Fq(\022)h Fv(M)1665 1627 y Fp(0)1716 1663 y Fq(\022)f Fv(M)45 b Fs(b)-5 b(e)33 b(submo)-5 b(dules)33 b(and)g(let)h Fv(N)44 b Fs(b)-5 b(e)33 b(lar)-5 b(ge)33 b(in)h Fv(M)10 b Fs(.)45 b(Then)315 1779 y Fv(N)403 1743 y Fp(0)461 1779 y Fs(is)35 b(lar)-5 b(ge)34 b(in)h Fv(M)1025 1743 y Fp(0)1049 1779 y Fs(.)148 1895 y Fx(\(2\))42 b Fs(L)-5 b(et)36 b Fv(N)41 b Fq(\022)32 b Fv(N)800 1859 y Fp(0)854 1895 y Fq(\022)f Fv(M)1066 1859 y Fp(0)1121 1895 y Fq(\022)g Fv(M)47 b Fs(b)-5 b(e)36 b(submo)-5 b(dules)36 b(and)g(let)g Fv(N)2418 1859 y Fp(0)2479 1895 y Fs(b)-5 b(e)36 b(smal)5 b(l)35 b(in)h Fv(M)3085 1859 y Fp(0)3110 1895 y Fs(.)49 b(Then)36 b Fv(N)47 b Fs(is)36 b(smal)5 b(l)315 2011 y(in)34 b Fv(M)10 b Fs(.)148 2128 y Fx(\(3\))42 b Fs(L)-5 b(et)35 b Fv(N)5 b(;)17 b(N)698 2091 y Fp(0)749 2128 y Fq(\022)28 b Fv(M)46 b Fs(b)-5 b(e)34 b(lar)-5 b(ge)35 b(submo)-5 b(dules)34 b(in)g Fv(M)10 b Fs(.)46 b(Then)34 b Fv(N)f Fq(\\)22 b Fv(N)2697 2091 y Fp(0)2756 2128 y Fs(is)35 b(lar)-5 b(ge)34 b(in)g Fv(M)10 b Fs(.)148 2244 y Fx(\(4\))42 b Fs(L)-5 b(et)35 b Fv(N)5 b(;)17 b(N)698 2208 y Fp(0)749 2244 y Fq(\022)28 b Fv(M)46 b Fs(b)-5 b(e)34 b(smal)5 b(l)34 b(submo)-5 b(dules)35 b(in)f Fv(M)10 b Fs(.)46 b(Then)34 b Fv(N)e Fx(+)22 b Fv(N)2730 2208 y Fp(0)2789 2244 y Fs(is)35 b(smal)5 b(l)34 b(in)g Fv(M)10 b Fs(.)0 2405 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))32 b(Let)h Fv(U)38 b Fq(\022)29 b Fv(M)944 2369 y Fp(0)1000 2405 y Fx(with)k Fv(N)1310 2369 y Fp(0)1356 2405 y Fq(\\)23 b Fv(U)38 b Fx(=)28 b(0.)43 b(Then)34 b Fv(N)e Fq(\\)23 b Fv(U)38 b Fx(=)28 b(0)k(hence)i Fv(U)k Fx(=)28 b(0.)0 2521 y(\(2\))f(Let)h Fv(U)38 b Fq(\022)29 b Fv(M)38 b Fx(with)28 b Fv(N)23 b Fx(+)12 b Fv(U)38 b Fx(=)27 b Fv(M)10 b Fx(,)30 b(then)e Fv(N)1743 2485 y Fp(0)1779 2521 y Fx(+)12 b Fv(U)38 b Fx(=)28 b Fv(M)10 b Fx(.)42 b(F)-8 b(rom)28 b Fv(N)2588 2485 y Fp(0)2623 2521 y Fx(+)12 b(\()p Fv(U)23 b Fq(\\)12 b Fv(M)3020 2485 y Fp(0)3044 2521 y Fx(\))28 b(=)g(\()p Fv(N)3340 2485 y Fp(0)3375 2521 y Fx(+)12 b Fv(U)e Fx(\))i Fq(\\)g Fv(M)3771 2485 y Fp(0)3824 2521 y Fx(=)0 2637 y Fv(M)37 b Fq(\\)27 b Fv(M)328 2601 y Fp(0)390 2637 y Fx(=)38 b Fv(M)608 2601 y Fp(0)670 2637 y Fx(w)m(e)i(get)f Fv(U)e Fq(\\)26 b Fv(M)1288 2601 y Fp(0)1350 2637 y Fx(=)38 b Fv(M)1568 2601 y Fp(0)1631 2637 y Fx(and)h(th)m(us)g Fv(M)2151 2601 y Fp(0)2213 2637 y Fq(\022)g Fv(U)49 b Fx(whic)m(h)40 b(implies)h Fv(N)48 b Fq(\022)39 b Fv(U)10 b Fx(.)62 b(No)m(w)39 b(from)0 2753 y Fv(N)33 b Fx(+)22 b Fv(U)38 b Fx(=)28 b Fv(M)43 b Fx(w)m(e)33 b(get)g Fv(U)38 b Fx(=)28 b Fv(M)10 b Fx(.)0 2870 y(\(3\))32 b(Let)h(\()p Fv(N)g Fq(\\)22 b Fv(N)657 2833 y Fp(0)681 2870 y Fx(\))g Fq(\\)h Fv(U)38 b Fx(=)27 b(0.)44 b(Then)33 b Fv(N)g Fq(\\)23 b Fx(\()p Fv(N)1737 2833 y Fp(0)1782 2870 y Fq(\\)g Fv(U)10 b Fx(\))28 b(=)g(0)k(hence)i Fv(N)2557 2833 y Fp(0)2603 2870 y Fq(\\)23 b Fv(U)38 b Fx(=)27 b(0)33 b(and)f(th)m(us)i Fv(U)k Fx(=)28 b(0.)0 2986 y(\(4\))d(Let)g(\() p Fv(N)17 b Fx(+)7 b Fv(N)621 2950 y Fp(0)645 2986 y Fx(\))g(+)g Fv(U)38 b Fx(=)27 b Fv(M)10 b Fx(.)42 b(Then)26 b Fv(N)18 b Fx(+)7 b(\()p Fv(N)1705 2950 y Fp(0)1735 2986 y Fx(+)g Fv(U)j Fx(\))28 b(=)f Fv(M)36 b Fx(hence)27 b Fv(N)2545 2950 y Fp(0)2575 2986 y Fx(+)7 b Fv(U)39 b Fx(=)27 b Fv(M)36 b Fx(and)25 b(th)m(us)h Fv(U)39 b Fx(=)27 b Fv(M)10 b Fx(.)99 b Fo(\003)0 3147 y Ft(Lemma)39 b(9.4.)j Fs(L)-5 b(et)35 b Fv(N)5 b(;)17 b(U)38 b Fq(\022)28 b Fv(M)46 b Fs(b)-5 b(e)35 b(submo)-5 b(dules.)148 3283 y Fx(\(1\))42 b Fs(If)48 b Fv(N)60 b Fs(is)48 b(maximal)g(w.r.t.)88 b(the)49 b(c)-5 b(ondition)48 b Fv(N)43 b Fq(\\)33 b Fv(U)65 b Fx(=)53 b(0)c Fs(then)g Fv(N)43 b Fx(+)33 b Fv(U)65 b Fq(\022)54 b Fv(M)60 b Fs(is)49 b(a)f(lar)-5 b(ge)315 3399 y(submo)g(dule.)148 3516 y Fx(\(2\))42 b Fs(If)j Fv(N)57 b Fs(is)46 b(minimal)f(w.r.t.)79 b(the)46 b(c)-5 b(ondition)45 b Fv(N)c Fx(+)30 b Fv(U)60 b Fx(=)48 b Fv(M)57 b Fs(then)46 b Fv(N)41 b Fq(\\)31 b Fv(U)60 b Fq(\022)49 b Fv(M)57 b Fs(is)46 b(a)g(smal)5 b(l)315 3632 y(submo)-5 b(dule.)148 3748 y Fx(\(3\))42 b Fs(Ther)-5 b(e)34 b(is)g(a)h(submo)-5 b(dule)34 b Fv(N)46 b Fs(that)35 b(is)g(maximal)e(w.r.t.)45 b Fv(N)32 b Fq(\\)23 b Fv(U)38 b Fx(=)28 b(0)p Fs(.)0 3909 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))48 b(Let)h Fv(V)76 b Fq(\022)55 b Fv(M)k Fx(with)49 b(\()p Fv(N)44 b Fx(+)33 b Fv(U)10 b Fx(\))33 b Fq(\\)h Fv(V)76 b Fx(=)54 b(0)49 b(b)s(e)f(giv)m(en.)93 b(W)-8 b(e)48 b(ha)m(v)m(e)i Fv(N)44 b Fq(\\)33 b Fv(U)65 b Fx(=)55 b(0.)91 b(Let)0 4025 y Fv(n)9 b Fx(+)g Fv(v)32 b Fx(=)c Fv(u)f Fq(2)h Fx(\()p Fv(N)19 b Fx(+)9 b Fv(V)22 b Fx(\))9 b Fq(\\)g Fv(U)h Fx(.)43 b(This)27 b(implies)h Fv(v)k Fx(=)27 b Fv(u)9 b Fq(\000)g Fv(n)28 b Fq(2)h Fx(\()p Fv(N)19 b Fx(+)9 b Fv(U)h Fx(\))f Fq(\\)g Fv(V)51 b Fx(=)28 b(0)e(hence)h Fv(n)h Fx(=)g Fv(u)f Fq(2)h Fv(N)20 b Fq(\\)9 b Fv(U)39 b Fx(=)27 b(0)0 4141 y(and)j(\()p Fv(N)c Fx(+)16 b Fv(V)21 b Fx(\))16 b Fq(\\)g Fv(U)38 b Fx(=)27 b(0.)43 b(Th)m(us)31 b Fv(N)26 b Fx(+)16 b Fv(V)49 b Fx(=)27 b Fv(N)10 b Fx(,)31 b(since)g Fv(N)39 b Fx(is)30 b(maximal)h(w.r.t.)42 b Fv(N)27 b Fq(\\)16 b Fv(U)38 b Fx(=)28 b(0.)42 b(This)30 b(implies)0 4258 y Fv(V)49 b Fq(\022)28 b Fv(N)43 b Fx(hence)34 b Fv(V)50 b Fq(\022)28 b Fx(\()p Fv(N)k Fx(+)22 b Fv(U)10 b Fx(\))23 b Fq(\\)g Fv(V)49 b Fx(=)27 b(0)33 b(and)f Fv(V)50 b Fx(=)27 b(0.)43 b(So)33 b(w)m(e)h(get)e(that)g Fv(N)h Fx(+)22 b Fv(U)38 b Fq(\022)29 b Fv(M)43 b Fx(is)33 b(large.)0 4374 y(\(2\))46 b(Let)h Fv(V)73 b Fq(\022)52 b Fv(M)58 b Fx(with)47 b(\()p Fv(N)42 b Fq(\\)33 b Fv(U)10 b Fx(\))32 b(+)f Fv(V)74 b Fx(=)51 b Fv(M)10 b Fx(.)87 b(W)-8 b(e)47 b(ha)m(v)m(e)h Fv(N)42 b Fx(+)32 b Fv(U)62 b Fx(=)51 b Fv(M)10 b Fx(.)87 b(Let)47 b Fv(m)52 b Fq(2)g Fv(M)57 b Fx(with)0 4490 y Fv(m)28 b Fx(=)g Fv(n)18 b Fx(+)h Fv(u)27 b Fq(2)h Fv(N)h Fx(+)19 b Fv(U)10 b Fx(.)43 b(F)-8 b(urthermore)32 b(let)f Fv(n)d Fx(=)f Fv(n)1863 4454 y Fp(0)1906 4490 y Fx(+)18 b Fv(v)35 b Fx(with)c Fv(n)2360 4454 y Fp(0)2411 4490 y Fq(2)d Fv(N)i Fq(\\)19 b Fv(U)41 b Fx(and)31 b Fv(v)h Fq(2)c Fv(V)52 b Fx(\(since)32 b Fv(n)c Fq(2)g Fv(M)10 b Fx(\).)0 4606 y(This)35 b(implies)g Fv(v)e Fq(2)d Fv(V)44 b Fq(\\)23 b Fv(N)45 b Fx(and)33 b Fv(m)d Fx(=)f(\()p Fv(n)1551 4570 y Fp(0)1598 4606 y Fx(+)22 b Fv(u)p Fx(\))h(+)f Fv(v)33 b Fq(2)d Fv(U)k Fx(+)22 b(\()p Fv(V)45 b Fq(\\)23 b Fv(N)10 b Fx(\))34 b(and)g(th)m(us)g(\()p Fv(N)g Fq(\\)23 b Fv(V)e Fx(\))i(+)g Fv(U)40 b Fx(=)29 b Fv(M)10 b Fx(.)0 4723 y(Since)36 b Fv(N)46 b Fx(is)36 b(minimal)g(w.r.t.)52 b Fv(N)34 b Fx(+)24 b Fv(U)43 b Fx(=)32 b Fv(M)46 b Fx(w)m(e)36 b(ha)m(v)m(e)g Fv(N)43 b Fx(=)32 b Fv(N)i Fq(\\)25 b Fv(V)57 b Fx(hence)36 b Fv(N)43 b Fq(\022)32 b Fv(V)22 b Fx(.)51 b(F)-8 b(rom)35 b(this)h(and)0 4839 y(from)c(\()p Fv(N)h Fq(\\)22 b Fv(U)10 b Fx(\))23 b(+)f Fv(V)49 b Fx(=)28 b Fv(M)43 b Fx(w)m(e)34 b(get)e Fv(V)50 b Fx(=)27 b Fv(M)10 b Fx(.)44 b(Th)m(us)35 b Fv(N)d Fq(\\)23 b Fv(U)38 b Fq(\022)28 b Fv(M)44 b Fx(is)33 b(small.)0 4955 y(\(3\))g(The)h(set)f Fq(V)k Fx(:=)29 b Fq(f)p Fv(V)50 b Fq(\022)29 b Fv(M)10 b Fq(j)p Fv(V)44 b Fq(\\)23 b Fv(U)39 b Fx(=)29 b(0)p Fq(g)j Fx(is)i(inductiv)m(ely)i(ordered,)e(for) e(let)i(\()p Fv(V)3034 4970 y Fr(i)3061 4955 y Fx(\))3099 4970 y Fr(i)p Fp(2)p Fr(I)3243 4955 y Fx(b)s(e)g(a)f(c)m(hain)g(in)h Fq(V)0 5071 y Fx(and)j(let)g Fv(x)e Fq(2)f Fx(\()p Fq([)p Fv(V)690 5086 y Fr(i)719 5071 y Fx(\))25 b Fq(\\)g Fv(U)10 b Fx(.)56 b(Then)38 b(there)f(is)g(an)g Fv(i)d Fq(2)h Fv(I)45 b Fx(with)37 b Fv(x)e Fq(2)g Fv(V)2516 5086 y Fr(i)2569 5071 y Fq(\\)25 b Fv(U)47 b Fx(hence)38 b Fv(x)d Fx(=)f(0.)56 b(Th)m(us)38 b Fq([)p Fv(V)3754 5086 y Fr(i)3819 5071 y Fx(in)0 5188 y Fq(V)i Fx(is)32 b(an)f(upp)s(er)h(b)s(ound)g(of)f (the)h Fv(V)1244 5203 y Fr(i)1272 5188 y Fx(.)43 b(Consequen)m(tly)34 b(there)f(is)f(a)f(submo)s(dule)i Fv(N)42 b Fx(of)31 b Fv(M)42 b Fx(that)31 b(is)h(maximal)0 5304 y(w.r.t.)44 b Fv(N)33 b Fq(\\)22 b Fv(U)39 b Fx(=)27 b(0.)3069 b Fo(\003)0 5465 y Ft(Lemma)39 b(9.5.)j Fv(N)c Fq(\022)28 b Fv(M)46 b Fs(is)35 b(lar)-5 b(ge)34 b(if)h(and)f(only)h(if)f(the)h (fol)5 b(lowing)34 b(holds)1149 5622 y Fq(8)p Fv(m)28 b Fq(2)g Fv(M)33 b Fq(n)22 b(f)p Fx(0)p Fq(g9)p Fv(r)30 b Fq(2)e Fv(R)h Fx(:)f Fv(r)s(m)f Fq(2)h Fv(N)33 b Fq(n)22 b(f)p Fx(0)p Fq(g)p Fv(:)p eop end %%Page: 77 77 TeXDict begin 77 76 bop 1650 -170 a Fu(Radical)26 b(and)f(So)r(cle)1574 b(77)0 29 y Fs(Pr)-5 b(o)g(of.)41 b Fv(N)47 b Fq(\022)36 b Fv(M)48 b Fx(large)38 b Fq(\()-17 b(\))36 b Fx([)p Fq(8)p Fv(U)47 b Fq(\022)36 b Fv(M)47 b Fx(:)36 b Fv(N)g Fq(\\)26 b Fv(U)47 b Fx(=)36 b(0)g(=)-17 b Fq(\))36 b Fv(U)46 b Fx(=)36 b(0])g Fq(\()-17 b(\))36 b Fx([)p Fq(8)p Fv(U)47 b Fq(\022)37 b Fv(M)46 b Fx(:)37 b Fv(U)46 b Fq(6)p Fx(=)36 b(0)g(=)-17 b Fq(\))0 187 y Fv(N)41 b Fq(\\)30 b Fv(U)58 b Fq(6)p Fx(=)48 b(0])632 122 y Fk(\()p Fp(\003)p Fk(\))586 187 y Fq(\()-17 b(\))47 b Fx([)p Fq(8)p Fv(R)q(m)h Fq(\022)g Fv(M)59 b Fx(:)47 b Fv(R)q(m)h Fq(6)p Fx(=)f(0)h(=)-17 b Fq(\))47 b Fv(N)41 b Fq(\\)30 b Fv(R)q(m)48 b Fq(6)p Fx(=)f(0])h Fq(\()-17 b(\))47 b Fx([)p Fq(8)p Fv(m)h Fq(2)g Fv(M)41 b Fq(n)30 b(f)p Fx(0)p Fq(g9)p Fv(r)50 b Fq(2)0 303 y Fv(R)41 b Fx(:)g Fv(r)s(m)f Fq(2)g Fv(N)e 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b Fq(g)36 b Fx(is)g(not)f(empt)m (y)i(since)f Fv(R)q(m)g Fx(is)g(not)f(small)h(in)f Fv(M)10 b Fx(.)52 b Fq(S)0 1125 y Fx(is)36 b(inductiv)m(ely)j(ordered.)53 b(In)36 b(fact)g(let)g(\()p Fv(U)1574 1140 y Fr(i)1602 1125 y Fq(j)p Fv(i)d Fq(2)g Fv(I)8 b Fx(\))36 b(b)s(e)f(a)h(c)m(hain)g (in)g Fq(S)7 b Fx(.)54 b(Then)37 b(w)m(e)f(ha)m(v)m(e)h Fv(m)d Fq(62)f Fv(U)3581 1140 y Fr(i)3645 1125 y Fx(for)i(all)0 1241 y Fv(i)c Fq(2)g Fv(I)8 b Fx(.)49 b(Hence)36 b Fq([)p Fv(U)712 1256 y Fr(i)771 1241 y Fn($)c Fv(M)45 b Fx(and)34 b(ob)m(viously)i Fv(R)q(m)24 b Fx(+)f(\()p Fq([)p Fv(U)2096 1256 y Fr(i)2125 1241 y Fx(\))31 b(=)g Fv(M)10 b Fx(.)49 b(Then)36 b(there)f(is)g(a)f(maximal)h(elemen)m(t)0 1357 y Fv(N)44 b Fx(in)33 b Fq(S)7 b Fx(.)46 b(Let)34 b Fv(N)39 b Fn($)29 b Fv(N)863 1321 y Fp(0)916 1357 y Fq(\022)g Fv(M)10 b Fx(.)46 b(Then)35 b Fv(R)q(m)23 b Fx(+)f Fv(N)1824 1321 y Fp(0)1877 1357 y Fx(=)28 b Fv(M)10 b Fx(.)46 b(Since)35 b Fv(N)2502 1321 y Fp(0)2566 1357 y Fv(=)-61 b Fq(2)30 b(S)40 b Fx(w)m(e)35 b(get)e Fv(N)3146 1321 y Fp(0)3199 1357 y Fx(=)28 b Fv(M)44 b Fx(hence)35 b Fv(N)43 b Fx(is)0 1474 y(a)32 b(maximal)i(submo)s(dule.)45 b(F)-8 b(urthermore)33 b(w)m(e)g(ha)m(v)m(e)h(ob)m(viously)g Fv(m)40 b(=)-61 b Fq(2)29 b Fv(N)10 b Fx(.)1127 b Fo(\003)0 1652 y Ft(De\014nition)38 b(9.7.)190 b Fx(\(1\))42 b(Radical\()p Fv(M)10 b Fx(\))28 b(=)f(Rad\()p Fv(M)10 b Fx(\))29 b(:=)e Fq(\\f)p Fv(U)39 b Fn($)28 b Fv(M)10 b Fq(j)p Fv(U)44 b Fx(maximal)33 b(submo)s(dule)p Fq(g)p Fx(,)148 1768 y(\(2\))42 b(So)s(cle\()p Fv(M)10 b Fx(\))28 b(=)g(So)s(c\()p Fv(M)10 b Fx(\))28 b(:=)1334 1694 y Fl(P)1439 1768 y Fq(f)p Fv(U)38 b Fq(\022)28 b Fv(M)10 b Fq(j)p Fv(U)44 b Fx(simple)34 b(submo)s(dule)p Fq(g)p Fx(.)0 1947 y Ft(Prop)s(osition)k(9.8.)190 b Fx(\(1\))41 b(Rad\()p Fv(M)10 b Fx(\))28 b(=)1623 1872 y Fl(P)1728 1947 y Fq(f)p Fv(V)49 b Fq(\022)28 b Fv(M)46 b Fs(smal)5 b(l)o Fq(g)p Fs(.)148 2063 y Fx(\(2\))42 b(So)s(c\()p Fv(M)10 b Fx(\))28 b(=)f Fq(\\f)p Fv(V)50 b Fq(\022)28 b Fv(M)46 b Fs(lar)-5 b(ge)o Fq(g)p Fs(.)0 2242 y(Pr)g(o)g(of.)41 b Fx(\(1\))28 b Fq(\023)p Fx(:)42 b(Let)28 b Fv(V)50 b Fq(\022)28 b Fv(M)39 b Fx(small.)k(F)-8 b(or)27 b(all)h(maximal)h (submo)s(dules)i Fv(U)38 b Fq(\022)28 b Fv(M)39 b Fx(w)m(e)30 b(ha)m(v)m(e)f Fv(U)39 b Fq(\022)28 b Fv(U)c Fx(+)13 b Fv(V)50 b Fn($)0 2358 y Fv(M)c Fx(since)36 b Fv(V)56 b Fx(is)36 b(small)f(and)g Fv(U)43 b Fq(6)p Fx(=)31 b Fv(M)10 b Fx(.)52 b(This)36 b(implies)g Fv(U)43 b Fx(=)31 b Fv(U)j Fx(+)24 b Fv(V)56 b Fx(and)36 b Fv(V)53 b Fq(\022)32 b Fv(U)10 b Fx(.)51 b(Th)m(us)37 b Fv(V)53 b Fq(\022)32 b(\\)p Fv(U)46 b Fx(and)0 2474 y(th)m(us)215 2400 y Fl(P)336 2474 y Fv(V)k Fq(\022)28 b(\\)p Fv(U)10 b Fx(.)0 2591 y Fq(\022)p Fx(:)42 b(If)29 b Fv(R)q(m)h Fx(is)g(not)f(small)g(in)h Fv(M)40 b Fx(then)30 b(b)m(y)g(9.6)e(there)i(is)g(a)f(maximal)h(submo)s (dule)h Fv(N)39 b Fx(in)30 b Fv(M)39 b Fx(with)30 b Fv(m)40 b(=)-61 b Fq(2)28 b Fv(N)10 b Fx(.)0 2707 y(So)36 b(w)m(e)i(ha)m(v)m(e) g Fv(m)46 b(=)-61 b Fq(2)35 b(\\)p 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Fr(i)3149 3172 y Fx(.)44 b(Let)33 b Fv(N)38 b Fq(\022)28 b(\\)p Fv(V)3739 3187 y Fr(i)3800 3172 y Fx(b)s(e)0 3288 y(giv)m(en.)48 b(Let)33 b Fv(X)42 b Fx(b)s(e)34 b(maximal)g(in)g Fv(M)44 b Fx(with)34 b Fv(N)g Fq(\\)23 b Fv(X)37 b Fx(=)29 b(0)34 b(\(Lemma)g(9.4)f(\(3\)\).)46 b(Then)35 b Fv(N)e Fx(+)23 b Fv(X)37 b Fx(=)29 b Fv(V)51 b Fq(\022)30 b Fv(M)0 3404 y Fx(is)35 b(large)f(b)m(y)i(Lemma)f(9.4)f(\(1\).)48 b(This)36 b(implies)g Fv(N)e Fx(+)23 b(\()p Fv(X)31 b Fq(\\)24 b Fx(\()p Fq(\\)p Fv(V)2357 3419 y Fr(i)2385 3404 y Fx(\)\))31 b(=)g(\()p Fv(N)j Fx(+)23 b Fv(X)8 b Fx(\))23 b Fq(\\)h Fx(\()p Fq(\\)p Fv(V)3249 3419 y Fr(i)3277 3404 y Fx(\))34 b(\(Lemma)h(9.2\))0 3520 y(=)28 b Fv(V)43 b Fq(\\)23 b Fx(\()p Fq(\\)p Fv(V)454 3535 y Fr(i)482 3520 y Fx(\))28 b(=)f Fq(\\)p Fv(V)774 3535 y Fr(i)835 3520 y Fx(and)33 b Fv(N)g Fq(\\)22 b Fx(\()p Fv(X)30 b Fq(\\)23 b Fx(\()p Fq(\\)p Fv(V)1623 3535 y Fr(i)1651 3520 y Fx(\)\))28 b(=)f(0.)43 b(So)33 b(w)m(e)g(ha)m(v)m(e)h Fv(N)f Fq(\010)23 b 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Fx(Rad\()p Fv(M)10 b Fx(\)\))58 b(=)g(Rad\()p Fv(M)10 b Fx(\))p Fv(=)17 b Fx(Rad)o(\()p Fv(M)10 b Fx(\))59 b(=)f(0.)96 b(If)50 b(Rad\()p Fv(M)5 b(=U)10 b Fx(\))59 b(=)e(0)50 b(then)0 5443 y(Rad\()p Fv(M)10 b Fx(\))23 b(+)f Fv(U)5 b(=U)38 b Fx(=)27 b(0)33 b(and)f(th)m(us)i(Rad\()p Fv(M)10 b Fx(\))23 b(+)f Fv(U)38 b Fx(=)27 b Fv(U)44 b Fx(so)32 b(that)h(Rad\()p Fv(M)10 b Fx(\))28 b Fq(\022)g Fv(U)10 b Fx(.)819 b Fo(\003)0 5622 y Ft(Lemma)39 b(9.21.)j Fs(If)34 b Fx(So)s(c\()p Fv(M)10 b Fx(\))29 b(=)e Fv(M)46 b Fs(then)35 b Fx(Rad)o(\()p Fv(M)10 b Fx(\))29 b(=)e(0)p Fs(.)p eop end %%Page: 79 79 TeXDict begin 79 78 bop 1650 -170 a Fu(Radical)26 b(and)f(So)r(cle)1574 b(79)0 29 y Fs(Pr)-5 b(o)g(of.)41 b Fx(If)h(So)s(c\()p Fv(M)10 b Fx(\))45 b(=)e Fv(M)53 b Fx(holds)43 b(then)f Fv(M)53 b Fx(is)43 b(semisimple.)74 b(So)42 b(no)g(submo)s(dule)i(is)e (small)h(and)f(th)m(us)0 146 y(Rad\()p Fv(M)10 b Fx(\))28 b(=)g(0.)3260 b Fo(\003)0 324 y Ft(Lemma)39 b(9.22.)j Fs(L)-5 b(et)35 b Fv(M)46 b Fs(b)-5 b(e)34 b(A)n(rtinian.)45 b(Then)34 b(we)g(have)1270 487 y Fx(Rad)o(\()p Fv(M)10 b Fx(\))29 b(=)e(0)h Fq(\()-17 b(\))27 b Fx(So)s(c\()p Fv(M)10 b Fx(\))28 b(=)g Fv(M)5 b(:)0 664 y Fs(Pr)-5 b(o)g(of.)41 b Fx(Let)24 b Fv(M)34 b Fx(b)s(e)24 b(Artinian)g(and)g (Rad\()p Fv(M)10 b Fx(\))28 b(=)g(0.)40 b(Let)24 b Fv(U)38 b Fq(\022)28 b Fv(M)35 b Fx(and)23 b Fv(N)34 b Fx(b)s(e)24 b(minimal)h(with)f Fv(N)14 b Fx(+)t Fv(U)39 b Fx(=)27 b Fv(M)10 b Fx(.)0 781 y(By)36 b(9.4)e(\(2\))h(w)m(e)h(ha)m(v)m(e)h Fv(N)d Fq(\\)24 b Fv(U)43 b Fq(\022)32 b Fv(M)46 b Fx(small)36 b(so)f(that)g Fv(N)g Fq(\\)24 b Fv(U)43 b Fx(=)31 b(0.)51 b(Th)m(us)37 b Fv(U)46 b Fx(is)35 b(a)g(direct)h(summand)g(of)0 897 y Fv(M)10 b Fx(,)33 b Fv(M)44 b Fx(is)33 b(semisimple)i(and)e Fv(M)38 b Fx(=)28 b(So)s(c\()p Fv(M)10 b Fx(\).)2147 b Fo(\003)0 1075 y Ft(Prop)s(osition)38 b(9.23.)j Fs(The)35 b(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent)34 b(for)h Fv(M)10 b Fs(:)148 1214 y Fx(\(1\))42 b Fv(M)j Fs(is)35 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Fx(\(1)24 b(+)g Fv(i)p Fx(\))h(+)e Fv(I)41 b Fx(=)32 b Fv(R)37 b Fx(it)0 3515 y(follo)m(ws)c Fv(R)q Fx(\(1)22 b(+)g Fv(i)p Fx(\))28 b(=)g Fv(R)q Fx(.)43 b(Th)m(us)34 b(1)22 b(+)g Fv(i)33 b Fx(is)g(a)f(righ)m(t)h(unit.)0 3631 y(\(2\))48 b(=)-17 b Fq(\))45 b Fx(\(3\):)68 b(Let)45 b Fv(k)s Fx(\(1)31 b(+)f Fv(i)p Fx(\))49 b(=)g(1.)80 b(This)46 b(implies)h Fv(k)s(i)i Fx(=)g(1)30 b Fq(\000)h Fv(k)52 b Fq(2)d Fv(I)k Fx(and)45 b(th)m(us)h Fv(k)34 b Fq(\000)d Fx(1)48 b Fq(2)i Fv(I)8 b Fx(.)80 b(So)0 3747 y Fv(k)32 b Fx(=)d(1)22 b(+)h(\()p Fv(k)i Fq(\000)e Fx(1\))33 b(is)h(a)f(righ)m(t)h(unit.)45 b(Since)35 b Fv(k)h Fx(is)e(also)f(a)g(left)h(unit,)g(w)m(e)g(get)f(\(1)23 b(+)f Fv(i)p Fx(\))p Fv(k)32 b Fx(=)d(1,)k(so)h(that)f(1)22 b(+)h Fv(i)0 3863 y Fx(is)33 b(a)f(unit.)0 3979 y(\(3\))37 b(=)-17 b Fq(\))38 b Fx(\(4\):)53 b(Giv)m(en)39 b Fv(i)e Fq(2)g Fv(I)45 b Fx(and)38 b Fv(r)i Fq(2)c Fv(R)q Fx(.)60 b(Then)38 b(1)26 b(+)f Fv(r)s(i)38 b Fx(is)g(a)g(unit)g(with)g(in)m(v)m (erse)j(\(1)25 b(+)h Fv(r)s(i)p Fx(\))3498 3943 y Fp(\000)p Fk(1)3592 3979 y Fx(.)59 b(Since)0 4096 y(\(1)24 b Fq(\000)h Fv(i)p Fx(\(1)f(+)h Fv(r)s(i)p Fx(\))576 4060 y Fp(\000)p Fk(1)670 4096 y Fv(r)s Fx(\)\(1)e(+)i Fv(ir)s Fx(\))33 b(=)g(1)24 b(+)g Fv(ir)k Fq(\000)d Fv(i)p Fx(\(1)f(+)g Fv(r)s(i)p Fx(\))1968 4060 y Fp(\000)p Fk(1)2062 4096 y Fx(\()p Fv(r)j Fx(+)e Fv(r)s(ir)s Fx(\))32 b(=)i(1)24 b(+)g Fv(ir)j Fq(\000)e Fv(i)p Fx(\(1)g(+)f Fv(r)s(i)p Fx(\))3321 4060 y Fp(\000)p Fk(1)3415 4096 y Fx(\(1)g(+)g Fv(r)s(i)p Fx(\))p Fv(r)36 b Fx(=)0 4212 y(1)19 b(+)h Fv(ir)i Fq(\000)e Fv(ir)31 b Fx(=)d(1)j(and)g(symmetrically)j(\(1)19 b(+)h Fv(ir)s Fx(\)\(1)f Fq(\000)h Fv(i)p Fx(\(1)f(+)h Fv(r)s(i)p Fx(\))2344 4176 y Fp(\000)p Fk(1)2438 4212 y Fv(r)s Fx(\))27 b(=)h(1)j(w)m(e)h(get)g(that)f(1)19 b(+)g Fv(ir)35 b Fx(is)c(a)g(unit.)0 4328 y(If)f Fv(a)g Fx(is)g(a)f(unit)i(and)f Fv(i)d Fq(2)i Fv(I)8 b Fx(,)30 b Fv(r)g Fq(2)e Fv(R)j Fx(then)f Fv(a)16 b Fx(+)g Fv(ir)34 b Fx(is)c(a)g(unit,)h(since)g Fv(a)p Fx(\(1)16 b(+)g Fv(a)2703 4292 y Fp(\000)p Fk(1)2798 4328 y Fv(ir)s Fx(\))28 b(=)f(\()p Fv(a)16 b Fx(+)g Fv(ir)s Fx(\))31 b(is)f(a)g(pro)s(duct)0 4444 y(of)i(t)m(w)m(o)h(units)h(b)m(y)f Fv(a)726 4408 y Fp(\000)p Fk(1)821 4444 y Fv(i)28 b Fq(2)g Fv(I)8 b Fx(.)0 4561 y(If)94 4486 y Fl(P)199 4512 y Fr(n)199 4590 y(k)r Fk(=1)349 4561 y Fv(i)382 4576 y Fr(k)425 4561 y Fv(r)469 4576 y Fr(k)539 4561 y Fq(2)28 b Fv(I)8 b(R)31 b Fx(then)f(1)16 b(+)1165 4486 y Fl(P)1285 4561 y Fv(i)1318 4576 y Fr(k)1361 4561 y Fv(r)1405 4576 y Fr(k)1477 4561 y Fx(is)30 b(a)f(unit,)i(since)f(1)16 b(+)2273 4486 y Fl(P)2394 4561 y Fv(i)2427 4576 y Fr(k)2470 4561 y Fv(r)2514 4576 y Fr(k)2584 4561 y Fx(=)28 b(\(\(\(1)16 b(+)g Fv(i)2992 4576 y Fk(1)3030 4561 y Fv(r)3074 4576 y Fk(1)3114 4561 y Fx(\))g(+)g Fv(i)3293 4576 y Fk(2)3331 4561 y Fv(r)3375 4576 y Fk(2)3415 4561 y Fx(\))h Fv(:)g(:)g(:)d Fx(+)i Fv(i)3724 4576 y Fr(n)3771 4561 y Fv(r)3815 4576 y Fr(n)3862 4561 y Fx(\))0 4677 y(and)33 b(eac)m(h)g(of)f(the)h(brac)m(k)m(eted)i (terms)e(is)h(a)e(unit.)0 4793 y(\(4\))46 b(=)-17 b Fq(\))47 b Fx(\(5\):)71 b(Let)46 b Fv(M)58 b Fx(b)s(e)46 b(\014nitely)i (generated)g(and)e Fv(I)8 b(M)62 b Fx(=)52 b Fv(M)10 b Fx(.)86 b(Let)47 b Fv(t)f Fx(b)s(e)h(the)g(minimal)g(length)0 4909 y(of)c(a)h(system)h(of)f(generators)g(of)f Fv(M)58 b Fx(=)46 b Fv(R)q(m)1711 4924 y Fk(1)1781 4909 y Fx(+)29 b Fv(:)17 b(:)g(:)30 b Fx(+)f Fv(R)q(m)2296 4924 y Fr(t)2326 4909 y Fx(.)77 b(By)45 b Fv(I)8 b(M)57 b Fx(=)47 b Fv(M)54 b Fx(eac)m(h)45 b(elemen)m(t)g(in)f Fv(M)0 5026 y Fx(can)h(b)s(e)g (represen)m(ted)i(as)e(a)f(\014nite)h(sum)h(of)e(the)h(form)2116 4951 y Fl(P)2238 5026 y Fv(i)2271 4989 y Fp(0)2271 5050 y Fr(j)2308 5026 y Fv(m)2393 4989 y Fp(0)2393 5050 y Fr(j)2430 5026 y Fx(;)50 b(the)45 b Fv(m)2772 4989 y Fp(0)2772 5050 y Fr(j)2854 5026 y Fx(can)g(b)s(e)f(represen)m(ted)j(as) e(a)0 5161 y(linear)c(com)m(bination)g(of)e(the)i Fv(m)1221 5176 y Fr(i)1250 5161 y Fx(.)66 b(So)40 b(there)h(are)f(co)s(e\016cien) m(ts)j Fv(i)2448 5176 y Fr(k)2491 5161 y Fv(r)2535 5176 y Fr(k)2618 5161 y Fq(2)e Fv(I)48 b Fx(with)41 b Fv(m)3131 5176 y Fk(1)3211 5161 y Fx(=)3328 5086 y Fl(P)3433 5112 y Fr(t)3433 5190 y(k)r Fk(=1)3582 5161 y Fv(i)3615 5176 y Fr(k)3658 5161 y Fv(r)3702 5176 y Fr(k)3745 5161 y Fv(m)3830 5176 y Fr(k)3873 5161 y Fx(.)0 5287 y(This)49 b(implies)g(\(1)32 b Fq(\000)h Fv(i)846 5302 y Fk(1)885 5287 y Fv(r)929 5302 y Fk(1)969 5287 y Fx(\))p Fv(m)1092 5302 y Fk(1)1185 5287 y Fx(=)1314 5213 y Fl(P)1419 5239 y Fr(t)1419 5316 y(k)r Fk(=2)1568 5287 y Fv(i)1601 5302 y Fr(k)1644 5287 y Fv(r)1688 5302 y Fr(k)1731 5287 y Fv(m)1816 5302 y Fr(k)1859 5287 y Fx(.)88 b(Since)49 b(also)e(1)33 b Fq(\000)f Fv(i)2678 5302 y Fk(1)2718 5287 y Fv(r)2762 5302 y Fk(1)2849 5287 y Fx(is)48 b(a)f(unit,)52 b(w)m(e)d(get)e Fv(m)3731 5302 y Fk(1)3824 5287 y Fx(=)0 5339 y Fl(P)105 5366 y Fr(t)105 5443 y(k)r Fk(=2)238 5414 y Fx(\(1)24 b Fq(\000)h Fv(i)484 5429 y Fk(1)523 5414 y Fv(r)567 5429 y Fk(1)607 5414 y Fx(\))645 5378 y Fp(\000)p Fk(1)739 5414 y Fv(i)772 5429 y Fr(k)815 5414 y Fv(r)859 5429 y Fr(k)902 5414 y Fv(m)987 5429 y Fr(k)1063 5414 y Fq(2)33 b Fv(R)q(m)1322 5429 y Fk(2)1386 5414 y Fx(+)24 b Fv(:)17 b(:)g(:)24 b Fx(+)g Fv(R)q(m)1885 5429 y Fr(t)1951 5414 y Fx(a)35 b(con)m(tradiction)i(to)e(the)h (minimalit)m(y)h(of)e Fv(t)p Fx(.)53 b(So)36 b(w)m(e)0 5530 y(ha)m(v)m(e)e Fv(M)k Fx(=)28 b(0.)0 5646 y(\(5\))j(=)-17 b Fq(\))31 b Fx(\(6\):)43 b Fv(I)8 b(M)30 b Fx(+)20 b Fv(U)38 b Fx(=)28 b Fv(M)38 b Fx(=)-17 b Fq(\))28 b Fv(I)8 b Fx(\()p Fv(M)d(=U)10 b Fx(\))28 b(=)g(\()p Fv(I)8 b(M)30 b Fx(+)20 b Fv(U)10 b Fx(\))p Fv(=U)39 b Fx(=)27 b Fv(M)5 b(=U)39 b Fx(=)-17 b Fq(\))27 b Fv(M)5 b(=U)39 b Fx(=)27 b(0)h(=)-17 b Fq(\))27 b Fv(M)39 b Fx(=)27 b Fv(U)10 b Fx(.)p eop end %%Page: 80 80 TeXDict begin 80 79 bop 0 -170 a Fu(80)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fx(\(6\))32 b(=)-17 b Fq(\))33 b Fx(\(7\):)43 b Fv(I)8 b(M)43 b Fx(small)33 b(in)g Fv(M)38 b Fx(=)-17 b Fq(\))28 b Fv(I)8 b(M)38 b Fq(\022)28 b Fx(Rad\()p Fv(M)10 b Fx(\).)0 146 y(\(7\))32 b(=)-17 b Fq(\))33 b Fx(\(1\):)43 b Fv(M)38 b Fx(=)28 b Fv(R)g Fx(=)-17 b Fq(\))28 b Fv(I)8 b(R)29 b Fq(\022)f Fx(Rad\()1541 161 y Fr(R)1598 146 y Fv(R)q Fx(\).)2085 b Fo(\003)0 320 y Ft(Corollary)39 b(9.25.)j Fx(Rad\()984 335 y Fr(R)1042 320 y Fv(R)q Fx(\))27 b(=)h(Rad\()p Fv(R)1573 335 y Fr(R)1631 320 y Fx(\))p Fs(.)0 494 y(Pr)-5 b(o)g(of.)41 b Fx(Let)j Fv(I)53 b Fx(=)46 b(Rad)o(\()914 509 y Fr(R)972 494 y Fv(R)q Fx(\).)75 b(Then)45 b(1)29 b(+)g Fv(I)51 b Fx(consists)45 b(of)e(units.)76 b(Since)44 b Fv(I)51 b Fx(is)44 b(a)f(righ)m(t)g(ideal,)k(w)m(e)d(get)0 611 y Fv(I)36 b Fq(\022)28 b Fx(Rad)o(\()p Fv(R)470 626 y Fr(R)528 611 y Fx(\).)44 b(By)33 b(symmetry)i(w)m(e)e(get)g(Rad\()1765 626 y Fr(R)1822 611 y Fv(R)q Fx(\))28 b(=)f(Rad\()p Fv(R)2353 626 y Fr(R)2411 611 y Fx(\).)1347 b Fo(\003)0 785 y Ft(Lemma)39 b(9.26.)j Fv(R)36 b Fs(left)f(A)n(rtinian)f Fx(=)-17 b Fq(\))28 b Fv(R)q(=)17 b Fx(Rad)o(\()p Fv(R)q Fx(\))35 b Fs(semisimple.)0 959 y(Pr)-5 b(o)g(of.)41 b Fx(By)26 b(8.12)g Fv(R)q(=)17 b Fx(Rad)o(\()p Fv(R)q Fx(\))25 b(is)i(Artinian.)41 b(By)27 b(9.20)e(Rad)o(\()p Fv(R)q(=)17 b Fx(Rad\()p Fv(R)q Fx(\)\))27 b(=)h(0)d(and)h(b)m(y)h(9.23)e Fv(R)q(=)17 b Fx(Rad)o(\()p Fv(R)q Fx(\))0 1076 y(is)33 b(semisimple.)3239 b Fo(\003)0 1250 y Ft(Lemma)39 b(9.27.)j Fv(R)36 b Fs(A)n(rtinian)e Fx(=)-17 b Fq(\))28 b Fx(Rad\()p Fv(R)q Fx(\))35 b Fs(nilp)-5 b(otent.)0 1424 y(Pr)g(o)g(of.)41 b Fx(Let)f Fv(I)48 b Fx(:=)40 b(Rad\()p Fv(R)q Fx(\).)65 b(Since)41 b Fv(R)g Fx(is)f(Artinian,)i(the)e(c)m(hain)h Fv(I)48 b Fq(\023)40 b Fv(I)2742 1388 y Fk(2)2822 1424 y Fq(\023)g Fv(I)2990 1388 y Fk(3)3069 1424 y Fq(\023)h Fv(:)17 b(:)g(:)39 b Fq(\023)i Fv(I)3510 1388 y Fr(t)p Fk(+1)3670 1424 y Fx(=)f Fv(:)17 b(:)g(:)0 1540 y Fx(b)s(ecomes)38 b(stationary)-8 b(.)56 b(Assume)38 b Fv(I)1314 1504 y Fr(t)1378 1540 y Fq(6)p Fx(=)d(0.)55 b(Since)38 b(also)f Fv(I)2130 1504 y Fr(t)2159 1540 y Fv(I)42 b Fq(6)p Fx(=)35 b(0)h(there)h(is)h(a)e(minimal)h(mo)s(dule)g Fv(K)42 b Fq(\022)35 b Fv(I)0 1657 y Fx(w.r.t.)53 b Fv(I)331 1621 y Fr(t)361 1657 y Fv(K)40 b Fq(6)p Fx(=)33 b(0.)52 b(So)36 b(there)g(exists)h(an)f Fv(x)d Fq(2)h Fv(K)42 b Fx(with)37 b Fv(I)2113 1621 y Fr(t)2142 1657 y Fv(x)c Fq(6)p Fx(=)g(0,)j(i.e.)54 b(w)m(e)36 b(ha)m(v)m(e)h Fv(K)j Fx(=)33 b Fv(R)q(x)p Fx(.)53 b(Because)37 b(of)0 1773 y Fv(I)51 1737 y Fr(t)80 1773 y Fv(K)f Fx(=)27 b Fv(I)353 1737 y Fr(t)p Fk(+1)473 1773 y Fv(K)35 b Fx(=)28 b Fv(I)746 1737 y Fr(t)776 1773 y Fx(\()p Fv(I)8 b(K)f Fx(\))27 b Fq(6)p Fx(=)h(0)33 b(and)g Fv(I)8 b(K)35 b Fq(\022)28 b Fv(K)40 b Fx(w)m(e)34 b(get)e Fv(I)8 b(K)35 b Fx(=)28 b Fv(K)7 b Fx(.)44 b(By)34 b(the)f(Lemma)g(of)g(Nak)-5 b(a)m(y)m(ama)33 b(w)m(e)0 1889 y(get)g Fv(K)h Fx(=)28 b(0,)k(a)h(con)m(tradiction,)g(so)g Fv(I)1365 1853 y Fr(t)1422 1889 y Fx(=)28 b(0.)2221 b Fo(\003)0 2063 y Ft(Theorem)38 b(9.28.)k Fx(\(Hopkins\))36 b Fs(L)-5 b(et)1373 2078 y Fr(R)1431 2063 y Fv(R)36 b Fs(b)-5 b(e)34 b(A)n(rtinian.)45 b(Then)2352 2078 y Fr(R)2410 2063 y Fv(R)35 b Fs(is)g(No)-5 b(etherian.)0 2238 y(Pr)g(o)g(of.)41 b Fx(Let)33 b Fv(I)i Fx(:=)28 b(Rad\()p Fv(R)q Fx(\))k(and)g Fv(I)1280 2202 y Fr(n)p Fk(+1)1445 2238 y Fx(=)27 b(0.)44 b(Then)33 b Fv(I)1973 2202 y Fr(i)2001 2238 y Fv(=I)2101 2202 y Fr(i)p Fk(+1)2251 2238 y Fx(is)g(an)f Fv(R)q(=I)8 b Fx(-mo)s(dule)33 b(and)f(it)g(is)h(Artinian)g(as)0 2354 y(an)g Fv(R)q Fx(-mo)s(dule.)47 b(So)33 b Fv(I)816 2318 y Fr(i)844 2354 y Fv(=I)944 2318 y Fr(i)p Fk(+1)1096 2354 y Fx(is)h(also)g (Artinian)f(as)h Fv(R)q(=I)8 b Fx(-mo)s(dule.)46 b(By)34 b(9.26)f Fv(R)q(=I)41 b Fx(is)34 b(semisimple)j(hence)0 2470 y Fv(I)51 2434 y Fr(i)79 2470 y Fv(=I)179 2434 y Fr(i)p Fk(+1)322 2470 y Fx(is)25 b(also)h(semisimple,)j(i.e.)41 b Fv(I)1332 2434 y Fr(i)1360 2470 y Fv(=I)1460 2434 y Fr(i)p Fk(+1)1606 2470 y Fx(=)28 b Fq(\010)1787 2485 y Fr(k)r Fp(2)p Fr(X)1940 2470 y Fv(E)2012 2485 y Fr(k)2080 2470 y Fx(with)d(simple)i Fv(R)q(=I)8 b Fx(-mo)s(dules)25 b Fv(E)3245 2485 y Fr(k)3288 2470 y Fx(.)41 b(Since)26 b Fv(I)3654 2434 y Fr(i)3682 2470 y Fv(=I)3782 2434 y Fr(i)p Fk(+1)0 2587 y Fx(is)34 b(Artinian)g(the)g(direct)h(sum)g(is)f (\014nite)g(hence)i Fv(I)1821 2550 y Fr(i)1849 2587 y Fv(=I)1949 2550 y Fr(i)p Fk(+1)2100 2587 y Fx(are)e(No)s(etherian)g (\(as)g Fv(R)q(=I)8 b Fx(-mo)s(dule)34 b(and)f(as)h Fv(R)q Fx(-)0 2703 y(mo)s(dule\).)48 b(With)34 b(the)g(exact)h(sequences)i(0) 30 b Fq(\000)-60 b(!)29 b Fv(I)1825 2667 y Fr(i)p Fk(+1)1973 2703 y Fq(\000)-59 b(!)29 b Fv(I)2171 2667 y Fr(i)2229 2703 y Fq(\000)-59 b(!)29 b Fv(I)2427 2667 y Fr(i)2455 2703 y Fv(=I)2555 2667 y Fr(i)p Fk(+1)2703 2703 y Fq(\000)-60 b(!)30 b Fx(0,)k(with)g Fv(I)3234 2667 y Fr(n)p Fk(+1)3401 2703 y Fx(=)c(0)p Fv(;)17 b(I)3651 2667 y Fk(0)3719 2703 y Fx(=)30 b Fv(R)0 2819 y Fx(and)j(with)g(8.25)f(w)m(e)h(get)g(b)m(y)g (induction)h(that)e Fv(R)i Fx(is)f(No)s(etherian.)1415 b Fo(\003)0 2993 y Ft(Corollary)39 b(9.29.)j Fs(If)874 3008 y Fr(R)931 2993 y Fv(I)36 b Fq(\022)1115 3008 y Fr(R)1173 2993 y Fv(R)g Fs(is)e(nilp)-5 b(otent)35 b(then)f Fv(I)i Fq(\022)28 b Fx(Rad\()p Fv(R)q Fx(\))p Fs(.)0 3168 y(Pr)-5 b(o)g(of.)41 b Fx(Let)36 b Fv(I)527 3132 y Fr(n)608 3168 y Fx(=)d(0)j(and)g Fv(i)d Fq(2)h Fv(I)8 b Fx(.)54 b(Then)37 b(\(1)24 b(+)g Fv(i)p Fx(\))h Fq(\001)f Fx(\(1)g Fq(\000)h Fv(i)g Fx(+)f Fv(i)2314 3132 y Fk(2)2378 3168 y Fq(\000)h Fv(:)17 b(:)g(:)24 b Fq(\006)h Fv(i)2754 3132 y Fr(n)p Fk(+1)2891 3168 y Fx(\))34 b(=)f(1)j(hence)h(\(1)24 b(+)h Fv(i)p Fx(\))36 b(is)g(a)0 3284 y(unit.)44 b(By)33 b(the)g(Lemma)g(of)f(Nak)-5 b(a)m(y)m(ama)33 b(w)m(e)h(get)f Fv(I)i Fq(\022)28 b Fx(Rad\()p Fv(R)q Fx(\).)1482 b Fo(\003)0 3458 y Ft(Prop)s(osition)38 b(9.30.)877 3473 y Fr(R)935 3458 y Fv(M)46 b Fs(is)34 b(\014nitely)h(gener)-5 b(ate)g(d)34 b(if)h(and)f(only)h(if)148 3595 y Fx(\(1\))42 b(Rad)o(\()p Fv(M)10 b Fx(\))29 b Fq(\022)f Fv(M)45 b Fs(is)35 b(smal)5 b(l,)34 b(and)148 3711 y Fx(\(2\))42 b Fv(M)5 b(=)17 b Fx(Rad)o(\()p Fv(M)10 b Fx(\))36 b Fs(is)e(\014nitely)h(gener)-5 b(ate)g(d.)0 3885 y(Pr)g(o)g(of.)41 b Fx(=)-17 b Fq(\))p Fx(:)43 b(trivial)33 b(b)m(y)h(9.12)e(.)0 4002 y Fq(\()-17 b Fx(=:)50 b(Let)36 b Fq(f)p 464 3947 56 4 v Fv(x)519 4017 y Fr(i)580 4002 y Fx(=)d Fv(x)744 4017 y Fr(i)797 4002 y Fx(+)24 b(Rad\()p Fv(M)10 b Fx(\))p Fq(j)p Fv(i)33 b Fx(=)g(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(n)p Fq(g)35 b Fx(b)s(e)h(a)g(set)g(of)f(generating)h(elemen)m(ts)i(of)e Fv(M)5 b(=)17 b Fx(Rad)o(\()p Fv(M)10 b Fx(\).)0 4118 y(Then)32 b Fv(M)38 b Fx(=)28 b Fv(R)q(x)619 4133 y Fk(1)677 4118 y Fx(+)17 b Fv(:)g(:)g(:)h Fx(+)g Fv(R)q(x)1127 4133 y Fr(n)1192 4118 y Fx(+)g(Rad\()p Fv(M)10 b Fx(\))31 b(whic)m(h)h(implies)g(b)m(y)g(\(1\))e(that)g Fv(M)39 b Fx(=)27 b Fv(R)q(x)3142 4133 y Fk(1)3200 4118 y Fx(+)18 b Fv(:)f(:)g(:)h Fx(+)f Fv(R)q(x)3650 4133 y Fr(n)3698 4118 y Fx(.)98 b Fo(\003)0 4292 y Ft(Corollary)47 b(9.31.)e Fv(M)51 b Fs(is)41 b(No)-5 b(etherian)40 b(if)g(and)h(only)f(if)h(for)f (al)5 b(l)40 b(submo)-5 b(dules)40 b Fv(U)49 b Fq(\022)39 b Fv(M)52 b Fs(the)40 b(fol)5 b(lowing)0 4408 y(hold:)148 4545 y Fx(\(1\))42 b(Rad)o(\()p Fv(U)10 b Fx(\))29 b Fq(\022)f Fv(U)45 b Fs(is)35 b(smal)5 b(l.)148 4661 y Fx(\(2\))42 b Fv(U)5 b(=)17 b Fx(Rad)o(\()p Fv(U)10 b Fx(\))35 b Fs(is)g(\014nitely)g(gener)-5 b(ate)g(d)p eop end %%Page: 81 81 TeXDict begin 81 80 bop 1746 -170 a Fu(Lo)r(calization)1671 b(81)1538 29 y Fx(10.)48 b Fw(Localiza)-7 b(tion)0 204 y Fx(10.1.)48 b Ft(Lo)s(cal)39 b(rings.)0 369 y(De\014nition)d(10.1.)41 b Fx(Let)32 b Fv(R)g Fx(b)s(e)f(a)g(ring.)43 b(An)31 b(elemen)m(t)i Fv(r)e Fq(2)d Fv(R)k Fx(is)g(called)g(a)e Fs(non)j(unit)p Fx(,)f(if)f Fv(r)j Fx(is)e(not)f(a)f(unit.)0 486 y(The)k(elemen)m(t)g Fv(r)h Fx(is)e(called)h Fs(invertible)p Fx(,)d(if)i Fv(r)i Fx(is)e(a)f(left)h(or)f(a)h(righ)m(t)f(unit.)0 602 y Fv(R)i Fx(is)f(called)g(a)f Fs(lo)-5 b(c)g(al)35 b(ring)p Fx(,)d(if)g(the)h(sum)h(of)e(an)m(y)h(t)m(w)m(o)g(non)g(in)m (v)m(ertible)i(elemen)m(ts)g(is)e(a)f(non)h(unit.)0 768 y Ft(Lemma)39 b(10.2.)j Fs(L)-5 b(et)35 b Fv(r)j Fs(b)-5 b(e)34 b(an)h(idemp)-5 b(otent)34 b Fx(\()p Fv(r)1761 731 y Fk(2)1828 768 y Fx(=)27 b Fv(r)s Fx(\))35 b Fs(in)f(a)h(lo)-5 b(c)g(al)34 b(ring)h Fv(R)q Fs(.)44 b(Then)34 b Fv(r)d Fx(=)c(0)35 b Fs(or)g Fv(r)30 b Fx(=)e(1)p Fs(.)0 933 y(Pr)-5 b(o)g(of.)41 b Fx(W)-8 b(e)36 b(ha)m(v)m(e)h(\(1)24 b Fq(\000)h Fv(r)s Fx(\))995 897 y Fk(2)1067 933 y Fx(=)33 b(1)24 b Fq(\000)g Fx(2)p Fv(r)j Fx(+)d Fv(r)1617 897 y Fk(2)1689 933 y Fx(=)33 b(1)24 b Fq(\000)h Fv(r)s Fx(.)52 b(Since)37 b(1)c(=)g(\(1)24 b Fq(\000)g Fv(r)s Fx(\))g(+)g Fv(r)39 b Fx(is)d(a)f(unit,)i Fv(r)h Fx(or)e(1)24 b Fq(\000)g Fv(r)0 1049 y Fx(is)37 b(in)m(v)m(ertible.)59 b(If)37 b Fv(r)i Fx(is)e(in)m(v)m(ertible,)k(e.g.)56 b(b)m(y)38 b Fv(sr)f Fx(=)e(1,)j(then)f(w)m(e)h(ha)m(v)m(e)g Fv(r)g Fx(=)d Fv(sr)2918 1013 y Fk(2)2992 1049 y Fx(=)g Fv(sr)i Fx(=)e(1.)56 b(If)37 b(1)24 b Fq(\000)i Fv(r)39 b Fx(is)0 1166 y(in)m(v)m(ertible)c(e.g.)44 b(b)m(y)33 b Fv(s)p Fx(\(1)22 b Fq(\000)g Fv(r)s Fx(\))28 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b(s)p Fx(\))28 b Fq(\030)g Fx(\()p Fv(r)1519 5605 y Fp(0)1542 5646 y Fv(;)17 b(s)1632 5605 y Fp(0)1655 5646 y Fx(\))27 b(:)p Fq(\()-17 b(\))60 b(9)p Fv(t)28 b Fq(2)g Fv(S)34 b Fx(:)28 b Fv(tsr)2479 5605 y Fp(0)2530 5646 y Fx(=)f Fv(ts)2714 5605 y Fp(0)2738 5646 y Fv(r)m(:)p eop end %%Page: 82 82 TeXDict begin 82 81 bop 0 -170 a Fu(82)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fv(R)q Fx([)p Fv(S)168 -7 y Fp(\000)p Fk(1)262 29 y Fx(])38 b(=)f Fv(S)506 -7 y Fp(\000)p Fk(1)600 29 y Fv(R)h Fx(:=)g Fv(R)27 b Fq(\002)f Fv(S=)38 b Fq(\030)g Fx(is)h(a)f(comm)m(utativ)m(e)i(ring)e(with)h (unit)g(elemen)m(t.)62 b(The)39 b(elemen)m(ts)i(are)0 146 y(denoted)34 b(b)m(y)1731 243 y Fv(r)p 1731 287 47 4 v 1731 378 a(s)1815 310 y Fx(:=)p 1946 223 207 4 v 28 w(\()p Fv(r)m(;)17 b(s)p Fx(\))o Fv(:)0 503 y Fx(The)34 b(map)1416 667 y Fv(')27 b Fx(:)h Fv(R)h Fq(3)f Fv(r)i Fq(7!)1971 600 y Fv(sr)p 1971 644 93 4 v 1994 735 a(s)2101 667 y Fq(2)e Fv(R)q Fx([)p Fv(S)2363 626 y Fp(\000)p Fk(1)2457 667 y Fx(])0 860 y(is)42 b(a)f(homomorphism)i(of)e(rings.)70 b(It)41 b(is)h(indep)s(enden)m(t)i(of)d(the)h(c)m(hoice)g(of)f Fv(s)i Fq(2)g Fv(S)6 b Fx(.)70 b(If)41 b Fv(R)h Fx(has)g(no)g(zero)0 976 y(divisors,)34 b(then)f Fv(')g Fx(is)g(injectiv)m(e.)0 1165 y Ft(Prop)s(osition)46 b(10.8.)g Fs(L)-5 b(et)43 b Fv(S)j Fq(\022)c Fv(R)h Fs(b)-5 b(e)42 b(a)g(multiplic)-5 b(atively)41 b(close)-5 b(d)41 b(set.)67 b(L)-5 b(et)2986 1180 y Fr(R)3044 1165 y Fv(M)52 b Fs(b)-5 b(e)42 b(an)g Fv(R)q Fs(-mo)-5 b(dule.)0 1282 y(Then)34 b(the)h(r)-5 b(elation)1024 1424 y Fx(\()p Fv(m;)17 b(s)p Fx(\))27 b Fq(\030)h Fx(\()p Fv(m)1530 1383 y Fp(0)1554 1424 y Fv(;)17 b(s)1644 1383 y Fp(0)1667 1424 y Fx(\))28 b(:)p Fq(\()-17 b(\))62 b(9)p Fv(t)28 b Fq(2)g Fv(S)34 b Fx(:)27 b Fv(tsm)2531 1383 y Fp(0)2583 1424 y Fx(=)g Fv(ts)2767 1383 y Fp(0)2791 1424 y Fv(m)0 1587 y Fs(on)34 b Fv(M)f Fq(\002)23 b Fv(S)41 b Fs(is)34 b(an)h(e)-5 b(quivalenc)g(e)33 b(r)-5 b(elation.)45 b(F)-7 b(urthermor)i(e)743 1793 y Fv(S)809 1751 y Fp(\000)p Fk(1)903 1793 y Fv(M)38 b Fx(:=)28 b Fv(M)33 b Fq(\002)22 b Fv(S=)28 b Fq(\030)163 b Fs(with)35 b(the)f(elements)2653 1725 y Fv(m)p 2653 1770 86 4 v 2673 1861 a(s)2776 1793 y Fx(:=)p 2907 1706 251 4 v 28 w(\()p Fv(m;)17 b(s)p Fx(\))0 2015 y Fs(is)35 b(an)f Fv(S)310 1979 y Fp(\000)p Fk(1)404 2015 y Fv(R)q Fs(-mo)-5 b(dule)34 b(with)h(the)g(op)-5 b(er)g(ations)1028 2183 y Fv(m)p 1028 2228 86 4 v 1048 2319 a(s)1146 2251 y Fx(+)1254 2183 y Fv(m)1339 2147 y Fp(0)p 1254 2228 109 4 v 1273 2319 a Fv(s)1319 2290 y Fp(0)1400 2251 y Fx(=)1514 2183 y Fv(s)1560 2147 y Fp(0)1583 2183 y Fv(m)22 b Fx(+)g Fv(sm)1919 2147 y Fp(0)p 1514 2228 430 4 v 1671 2319 a Fv(ss)1763 2290 y Fp(0)2087 2251 y Fs(and)2386 2183 y Fv(r)p 2386 2228 47 4 v 2386 2319 a(s)2453 2183 y(m)p 2453 2228 86 4 v 2472 2319 a(s)2548 2167 y Fp(0)2599 2251 y Fx(=)2713 2183 y Fv(r)s(m)p 2713 2228 132 4 v 2721 2319 a(ss)2813 2290 y Fp(0)2854 2251 y Fv(:)0 2479 y Fs(Pr)-5 b(o)g(of.)41 b Fx(as)33 b(in)g(Basic)g(Algebra)g(for)f Fv(S)1372 2443 y Fp(\000)p Fk(1)1466 2479 y Fv(R)q Fx(.)2255 b Fo(\003)0 2669 y Ft(Problem)38 b(10.1.)k Fx(Giv)m(e)33 b(a)f(complete)i(pro)s(of)e(of)g(Prop)s(osition)h(10.8.)0 2858 y Ft(Lemma)39 b(10.9.)679 2819 y Fr(m)p 679 2835 63 4 v 694 2893 a(s)779 2858 y Fx(=)28 b(0)34 b Fs(holds)g(in)h Fv(S)1401 2822 y Fp(\000)p Fk(1)1495 2858 y Fv(M)46 b Fs(if)34 b(and)h(only)f(if)h(ther)-5 b(e)35 b(is)f(a)h Fv(t)28 b Fq(2)g Fv(S)41 b Fs(with)34 b Fv(tm)29 b Fx(=)e(0)p Fs(.)0 3049 y(Pr)-5 b(o)g(of.)41 b Fx(\()p Fv(m;)17 b(s)p Fx(\))28 b Fq(\030)g Fx(\(0)p Fv(;)17 b(s)859 3013 y Fp(0)881 3049 y Fx(\))56 b Fq(\()-17 b(\))55 b(9)p Fv(t)1303 3013 y Fp(0)1354 3049 y Fq(2)28 b Fv(S)34 b Fx(:)28 b Fv(t)1632 3013 y Fp(0)1655 3049 y Fv(s)1701 3013 y Fp(0)1724 3049 y Fv(m)g Fx(=)g(0)55 b Fq(\()-17 b(\))55 b(9)p Fv(t)2373 3013 y Fp(0)2397 3049 y Fv(s)2443 3013 y Fp(0)2494 3049 y Fq(2)28 b Fv(S)33 b Fx(:)28 b Fv(t)2771 3013 y Fp(0)2795 3049 y Fv(s)2841 3013 y Fp(0)2864 3049 y Fv(m)g Fx(=)f(0.)667 b Fo(\003)0 3238 y Ft(Lemma)39 b(10.10.)190 b Fx(\(1\))42 b Fv(')1104 3253 y Fr(M)1210 3238 y Fx(:)28 b Fv(M)38 b Fq(3)29 b Fv(m)f Fq(7!)1742 3199 y Fr(sm)p 1742 3215 96 4 v 1773 3273 a(s)1874 3238 y Fq(2)h Fv(S)2035 3202 y Fp(\000)p Fk(1)2129 3238 y Fv(M)42 b Fs(is)31 b(a)g(homomorphism)e (of)h(gr)-5 b(oups)31 b(indep)-5 b(en-)315 3355 y(dent)34 b(of)h Fv(s)27 b Fq(2)i Fv(S)6 b Fs(.)148 3471 y Fx(\(2\))42 b Fv(')379 3486 y Fr(M)498 3471 y Fs(is)f(inje)-5 b(ctive)39 b(if)i(and)f(only)h(if)f Fv(S)47 b Fs(c)-5 b(ontains)39 b(no)i(zer)-5 b(o)40 b(divisors)g(for)g Fv(M)10 b Fs(,)43 b(i.e.)62 b Fv(sm)39 b Fx(=)f(0)g(=)-17 b Fq(\))315 3587 y Fv(m)28 b Fx(=)f(0)p Fs(.)148 3703 y Fx(\(3\))42 b Fv(')379 3718 y Fr(M)492 3703 y Fs(is)35 b(bije)-5 b(ctive)34 b(if)h(and)f(only)h(if)f(the)h(map)f Fv(M)39 b Fq(3)28 b Fv(m)g Fq(7!)f Fv(sm)h Fq(2)g Fv(M)46 b Fs(is)35 b(bije)-5 b(ctive)34 b(for)g(al)5 b(l)35 b Fv(s)27 b Fq(2)i Fv(S)6 b Fs(.)148 3820 y Fx(\(4\))42 b Fv(')379 3835 y Fr(R)471 3820 y Fs(is)35 b(a)f(homomorphism)f(of)i(rings.)148 3936 y Fx(\(5\))42 b Fv(')379 3951 y Fr(M)485 3936 y Fx(:)28 b Fv(M)38 b Fq(\000)-57 b(!)28 b Fv(S)886 3900 y Fp(\000)p Fk(1)980 3936 y Fv(M)45 b Fs(is)35 b Fv(')1288 3951 y Fr(R)1346 3936 y Fs(-semiline)-5 b(ar,)33 b(i.e.)44 b Fv(')2123 3951 y Fr(M)2202 3936 y Fx(\()p Fv(r)s(m)p Fx(\))28 b(=)f Fv(')2605 3951 y Fr(R)2663 3936 y Fx(\()p Fv(r)s Fx(\))p Fv(')2850 3951 y Fr(M)2928 3936 y Fx(\()p Fv(m)p Fx(\))p Fs(.)0 4125 y(Pr)-5 b(o)g(of.)41 b Fx(\(1\))32 b Fv(t)490 4089 y Fp(0)514 4125 y Fx(\()p Fv(tsm)22 b Fq(\000)h Fv(stm)p Fx(\))28 b(=)g(0)k(implies)1598 4086 y Fr(sm)p 1598 4102 V 1629 4160 a(s)1731 4125 y Fx(=)1844 4086 y Fr(tm)p 1844 4102 88 4 v 1875 4160 a(t)1942 4125 y Fx(.)0 4243 y(\(2\))g Fv(')221 4258 y Fr(M)300 4243 y Fx(\()p Fv(m)p Fx(\))c(=)f(0)56 b Fq(\()-17 b(\))945 4203 y Fr(sm)p 945 4220 96 4 v 976 4277 a(s)1077 4243 y Fx(=)28 b(0)55 b Fq(\()-17 b(\))55 b(9)p Fv(t)28 b Fq(2)g Fv(S)34 b Fx(:)27 b Fv(tm)i Fx(=)e(0)33 b(b)m(y)g(10.9.)0 4376 y(\(3\))h Fv(')223 4391 y Fr(M)336 4376 y Fx(surjectiv)m(e)67 b Fq(\()-17 b(\))61 b(8)1114 4336 y Fr(m)p 1114 4353 63 4 v 1129 4410 a(s)1218 4376 y Fq(2)31 b Fv(S)1381 4339 y Fp(\000)p Fk(1)1475 4376 y Fv(M)45 b Fq(9)p Fv(m)1754 4339 y Fp(0)1808 4376 y Fq(2)31 b Fv(M)42 b Fx(:)2109 4336 y Fr(sm)2204 4313 y Fh(0)p 2109 4353 118 4 v 2151 4410 a Fr(s)2267 4376 y Fx(=)2383 4336 y Fr(m)p 2383 4353 63 4 v 2398 4410 a(s)2517 4376 y Fq(\()-17 b(\))61 b(8)p Fv(m)31 b Fq(2)g Fv(M)5 b(;)17 b(s)31 b Fq(2)g Fv(S)40 b Fq(9)p Fv(m)3586 4339 y Fp(0)3640 4376 y Fq(2)31 b Fv(M)42 b Fx(:)0 4493 y Fv(sm)131 4457 y Fp(0)182 4493 y Fx(=)28 b Fv(m)55 b Fq(\()-17 b(\))55 b(8)p Fv(s)28 b Fq(2)h Fv(S)k Fx(:)28 b(\()p Fv(s)p Fq(\001)f Fx(:)h Fv(M)38 b Fq(\000)-59 b(!)27 b Fv(M)10 b Fx(\))33 b(surjectiv)m(e.)0 4621 y(\(4\))f(+)h(\(5\))f Fv(')487 4636 y Fr(M)566 4621 y Fx(\()p Fv(r)s(m)p Fx(\))27 b(=)915 4582 y Fr(s)948 4558 y Fg(2)982 4582 y Fr(r)r(m)p 915 4598 164 4 v 963 4655 a(s)996 4636 y Fg(2)1116 4621 y Fx(=)1229 4582 y Fr(sr)p 1229 4598 67 4 v 1246 4655 a(s)1316 4582 y(sm)p 1316 4598 96 4 v 1347 4655 a(s)1448 4621 y Fx(=)h Fv(')1616 4636 y Fr(R)1673 4621 y Fx(\()p Fv(r)s Fx(\))p Fv(')1860 4636 y Fr(M)1939 4621 y Fx(\()p Fv(m)p Fx(\))p Fv(:)1696 b Fo(\003)0 4816 y Ft(Corollary)39 b(10.11.)j Fv(S)893 4780 y Fp(\000)p Fk(1)1015 4816 y Fx(:)28 b Fv(R)q Fs(-)p Fx(Mo)s(d)g Fq(\000)-57 b(!)27 b Fv(S)1616 4780 y Fp(\000)p Fk(1)1710 4816 y Fv(R)q Fs(-)p Fx(Mo)s(d)35 b Fs(is)f(an)h(additive)f (functor.)0 5014 y(Pr)-5 b(o)g(of.)41 b Fx(F)-8 b(or)32 b Fv(f)39 b Fq(2)29 b Fx(Hom)858 5029 y Fr(R)916 5014 y Fx(\()p Fv(M)5 b(;)17 b(N)10 b Fx(\))33 b(w)m(e)h(form)e Fv(S)1696 4978 y Fp(\000)p Fk(1)1790 5014 y Fv(f)39 b Fq(2)29 b Fx(Hom)2175 5031 y Fr(S)2222 5012 y Fh(\000)p Fg(1)2304 5031 y Fr(R)2362 5014 y Fx(\()p Fv(S)2466 4978 y Fp(\000)p Fk(1)2560 5014 y Fv(M)5 b(;)17 b(S)2769 4978 y Fp(\000)p Fk(1)2863 5014 y Fv(N)10 b Fx(\))33 b(b)m(y)h Fv(S)3224 4978 y Fp(\000)p Fk(1)3318 5014 y Fv(f)11 b Fx(\()3425 4975 y Fr(m)p 3425 4991 63 4 v 3440 5048 a(s)3497 5014 y Fx(\))28 b(:=)3704 4967 y Fr(f)7 b Fk(\()p Fr(m)p Fk(\))p 3704 4991 159 4 v 3767 5048 a Fr(s)3873 5014 y Fx(.)0 5136 y(In)27 b(order)f(to)g(sho)m(w)h(that)f Fv(S)983 5099 y Fp(\000)p Fk(1)1077 5136 y Fv(f)37 b Fx(is)27 b(a)f(w)m(ell)h(de\014ned)h(map)f(assume)g(\()p Fv(m;)17 b(s)p Fx(\))28 b Fq(\030)g Fx(\()p Fv(m)2898 5099 y Fp(0)2922 5136 y Fv(;)17 b(s)3012 5099 y Fp(0)3035 5136 y Fx(\).)41 b(Then)27 b Fv(ts)3470 5099 y Fp(0)3494 5136 y Fv(m)h Fx(=)f Fv(tsm)3876 5099 y Fp(0)0 5267 y Fx(for)32 b(a)g Fv(t)c Fq(2)g Fv(S)39 b Fx(and)32 b(th)m(us)i Fv(ts)971 5231 y Fp(0)994 5267 y Fv(f)11 b Fx(\()p Fv(m)p Fx(\))28 b(=)g Fv(tsf)11 b Fx(\()p Fv(m)1609 5231 y Fp(0)1632 5267 y Fx(\).)43 b(This)34 b(implies)2304 5220 y Fr(f)7 b Fk(\()p Fr(m)p Fk(\))p 2304 5245 V 2367 5302 a Fr(s)2500 5267 y Fx(=)2614 5220 y Fr(f)g Fk(\()p Fr(m)2744 5197 y Fh(0)2767 5220 y Fk(\))p 2614 5245 181 4 v 2677 5302 a Fr(s)2710 5283 y Fh(0)2805 5267 y Fx(.)0 5389 y(With)50 b(the)h(usual)g(rules)g(for)e(calculations)j(with)e(fractions)h(one)f (pro)m(v)m(es)i(that)e Fv(S)3161 5353 y Fp(\000)p Fk(1)3255 5389 y Fv(f)61 b Fx(is)50 b(an)g Fv(S)3698 5353 y Fp(\000)p Fk(1)3792 5389 y Fv(R)q Fx(-)0 5505 y(homomorphism)43 b(and)e(that)h Fv(S)1183 5469 y Fp(\000)p Fk(1)1294 5505 y Fx(id)1375 5520 y Fr(M)1497 5505 y Fx(=)h(id)1697 5522 y Fr(S)1744 5503 y Fh(\000)p Fg(1)1826 5522 y Fr(M)1905 5505 y Fx(,)h Fv(S)2042 5469 y Fp(\000)p Fk(1)2136 5505 y Fx(\()p Fv(f)11 b(g)t Fx(\))42 b(=)h Fv(S)2549 5469 y Fp(\000)p Fk(1)2643 5505 y Fx(\()p Fv(f)11 b Fx(\))p Fv(S)2844 5469 y Fp(\000)p Fk(1)2937 5505 y Fx(\()p Fv(g)t Fx(\))41 b(and)h Fv(S)3370 5469 y Fp(\000)p Fk(1)3464 5505 y Fx(\()p Fv(f)d Fx(+)28 b Fv(g)t Fx(\))42 b(=)0 5622 y Fv(S)66 5585 y Fp(\000)p Fk(1)160 5622 y Fx(\()p Fv(f)11 b Fx(\))22 b(+)g Fv(S)481 5585 y Fp(\000)p Fk(1)575 5622 y Fx(\()p Fv(g)t Fx(\))32 b(hold.)2878 b Fo(\003)p eop end %%Page: 83 83 TeXDict begin 83 82 bop 1746 -170 a Fu(Lo)r(calization)1671 b(83)0 29 y Ft(Prop)s(osition)38 b(10.12.)j Fs(The)35 b(map)983 221 y Fv(\013)q Fx(\()p Fv(M)10 b Fx(\))28 b(:)g Fv(S)1375 180 y Fp(\000)p Fk(1)1469 221 y Fv(R)23 b Fq(\012)1643 236 y Fr(R)1723 221 y Fv(M)39 b Fq(3)1960 153 y Fv(r)p 1960 198 47 4 v 1960 289 a(s)2039 221 y Fq(\012)22 b Fv(m)28 b Fq(7!)2388 153 y Fv(r)s(m)p 2388 198 132 4 v 2432 289 a(s)2558 221 y Fq(2)g Fv(S)2718 180 y Fp(\000)p Fk(1)2812 221 y Fv(M)0 420 y Fs(de\014nes)34 b(a)g(functorial)h(isomorphism)1419 589 y Fv(\013)28 b Fx(:)g Fv(S)1630 548 y Fp(\000)p Fk(1)1724 589 y Fv(R)23 b Fq(\012)1898 604 y Fr(R)1979 589 y Fv(M)2111 562 y Fq(\030)2112 593 y Fx(=)2216 589 y Fv(S)2282 548 y Fp(\000)p Fk(1)2376 589 y Fv(M)0 759 y Fs(of)35 b(functors)f Fv(S)560 723 y Fp(\000)p Fk(1)655 759 y Fv(R)23 b Fq(\012)829 774 y Fr(R)909 759 y Fs(-)p Fv(;)52 b(S)1089 723 y Fp(\000)p Fk(1)1183 759 y Fs(-)27 b Fx(:)h Fv(R)q Fs(-)p Fx(Mo)s(d)g Fq(\000)-57 b(!)27 b Fv(S)1846 723 y Fp(\000)p Fk(1)1940 759 y Fv(R)q 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Fp(0)3152 1073 y Fv(m)33 b Fx(=)-17 b Fq(\))3472 1034 y Fr(r)r(m)p 3472 1050 97 4 v 3503 1108 a(s)3618 1073 y Fx(=)3744 1034 y Fr(r)3778 1011 y Fh(0)3801 1034 y Fr(m)p 3744 1050 119 4 v 3776 1108 a(s)3809 1089 y Fh(0)3873 1073 y Fx(.)0 1191 y(F)-8 b(urthermore)35 b Fl(e)-61 b Fv(\013)q Fx(\()p Fv(M)10 b Fx(\))29 b(is)g(ob)m(viously)h(additiv)m(e)g(in)f(b)s(oth)f (argumen)m(ts.)43 b(Finally)29 b(w)m(e)g(ha)m(v)m(e)37 b Fl(e)-62 b Fv(\013)q Fx(\()p Fv(M)10 b Fx(\)\()3550 1151 y Fr(r)p 3550 1168 34 4 v 3551 1225 a(s)3594 1191 y Fv(t;)17 b(m)p Fx(\))28 b(=)10 1274 y Fr(r)r(tm)p 10 1290 122 4 v 55 1348 a(s)169 1313 y Fx(=)34 b Fl(e)-61 b Fv(\013)q Fx(\()p Fv(M)10 b Fx(\)\()564 1274 y Fr(r)p 564 1290 34 4 v 565 1348 a(s)608 1313 y Fv(;)17 b(tm)p Fx(\),)33 b(i.e.)50 b Fl(e)-61 b Fv(\013)p Fx(\()p Fv(M)10 b Fx(\))34 b(is)f Fv(R)q Fx(-bilinear.)0 1436 y(W)-8 b(e)26 b(de\014ne)h(an)e(in)m(v)m(erse)j(map)e Fv(\014)6 b Fx(\()p Fv(M)k Fx(\))28 b(:)f Fv(S)1479 1400 y Fp(\000)p Fk(1)1573 1436 y Fv(M)39 b Fq(3)1810 1397 y Fr(m)p 1810 1413 63 4 v 1825 1470 a(s)1910 1436 y Fq(7!)2064 1397 y Fr(t)p 2047 1413 59 4 v 2047 1470 a(st)2123 1436 y Fq(\012)8 b Fv(m)28 b Fq(2)g Fv(S)2481 1400 y Fp(\000)p Fk(1)2575 1436 y Fv(R)8 b Fq(\012)2734 1451 y Fr(R)2801 1436 y Fv(M)i Fx(.)42 b(The)26 b(map)g Fv(\014)6 b Fx(\()p Fv(M)k Fx(\))26 b(is)g(w)m(ell)0 1569 y(de\014ned,)36 b(since)615 1530 y Fr(m)p 615 1546 63 4 v 630 1603 a(s)717 1569 y Fx(=)833 1530 y Fr(m)895 1506 y Fh(0)p 833 1546 85 4 v 848 1603 a Fr(s)881 1585 y Fh(0)960 1569 y Fx(=)-17 b Fq(\))32 b(9)p Fv(t)1241 1533 y Fp(0)1295 1569 y Fq(2)e Fv(S)36 b Fx(:)30 b Fv(t)1579 1533 y Fp(0)1602 1569 y Fv(s)1648 1533 y Fp(0)1672 1569 y Fv(m)g Fx(=)f Fv(t)1927 1533 y Fp(0)1951 1569 y Fv(sm)2082 1533 y Fp(0)2138 1569 y Fx(=)-17 b Fq(\))2356 1530 y Fr(t)p 2339 1546 59 4 v 2339 1603 a(st)2430 1569 y Fq(\012)24 b Fv(m)30 b Fx(=)2778 1530 y Fr(ts)2836 1506 y Fh(0)2858 1530 y Fr(t)2883 1506 y Fh(0)p 2762 1546 161 4 v 2762 1603 a Fr(sts)2853 1585 y Fh(0)2875 1603 y Fr(t)2900 1585 y Fh(0)2956 1569 y Fq(\012)23 b Fv(m)30 b Fx(=)3355 1530 y Fr(t)p 3287 1546 V 3287 1603 a(sts)3378 1585 y Fh(0)3400 1603 y Fr(t)3425 1585 y Fh(0)3481 1569 y Fq(\012)23 b Fv(s)3627 1533 y Fp(0)3650 1569 y Fv(t)3685 1533 y Fp(0)3709 1569 y Fv(m)30 b Fx(=)78 1663 y Fr(t)p 10 1679 V 10 1736 a(sts)101 1717 y Fh(0)123 1736 y Fr(t)148 1717 y Fh(0)203 1702 y Fq(\012)23 b Fv(st)384 1666 y Fp(0)407 1702 y Fv(m)28 b Fx(=)661 1663 y Fr(tst)744 1639 y Fh(0)p 634 1679 V 634 1736 a Fr(sts)725 1717 y Fh(0)747 1736 y Fr(t)772 1717 y Fh(0)827 1702 y Fq(\012)22 b Fv(m)1011 1666 y Fp(0)1063 1702 y Fx(=)1204 1663 y Fr(t)p 1176 1679 81 4 v 1176 1736 a(s)1209 1717 y Fh(0)1231 1736 y Fr(t)1289 1702 y Fq(\012)g Fv(m)1473 1666 y Fp(0)1497 1702 y Fx(.)0 1825 y(W)-8 b(e)33 b(ha)m(v)m(e)h Fv(\014)6 b(\013)28 b Fx(=)f(id)q(,)32 b(since)i Fv(\014)6 b Fx(\()p Fv(M)k Fx(\))p Fv(\013)q Fx(\()p Fv(M)g Fx(\)\()1559 1786 y Fr(r)p 1559 1802 34 4 v 1560 1859 a(s)1626 1825 y Fq(\012)22 b Fv(m)p Fx(\))28 b(=)g Fv(\014)6 b Fx(\()p Fv(M)k Fx(\)\()2269 1786 y Fr(r)r(m)p 2269 1802 97 4 v 2301 1859 a(s)2375 1825 y Fx(\))28 b(=)2571 1786 y Fr(t)p 2554 1802 59 4 v 2554 1859 a(st)2644 1825 y Fq(\012)23 b Fv(r)s(m)28 b Fx(=)3017 1786 y Fr(r)r(t)p 3017 1802 60 4 v 3018 1859 a(st)3109 1825 y Fq(\012)22 b Fv(m)28 b Fx(=)3435 1786 y Fr(r)p 3435 1802 34 4 v 3436 1859 a(s)3501 1825 y Fq(\012)22 b Fv(m)p Fx(.)0 1948 y(Similarly)34 b(w)m(e)f(ha)m(v)m(e)h Fv(\013)q(\014)f Fx(=)28 b(id,)33 b(since)h Fv(\013)q Fx(\()p Fv(M)10 b Fx(\))p Fv(\014)c Fx(\()p Fv(M)k Fx(\)\()1950 1908 y Fr(m)p 1950 1925 63 4 v 1965 1982 a(s)2022 1948 y Fx(\))28 b(=)g Fv(\013)q Fx(\()p Fv(M)10 b Fx(\)\()2499 1908 y Fr(t)p 2483 1925 59 4 v 2483 1982 a(st)2573 1948 y Fq(\012)23 b Fv(m)p Fx(\))28 b(=)2937 1908 y Fr(tm)p 2937 1925 88 4 v 2952 1982 a(st)3063 1948 y Fx(=)3176 1908 y Fr(m)p 3176 1925 63 4 v 3191 1982 a(s)3248 1948 y Fx(.)0 2081 y Fv(\013)43 b Fx(is)g(an)f Fv(S)424 2044 y Fp(\000)p Fk(1)518 2081 y Fv(R)q Fx(-homomorphism,)k(since)e Fv(\013)q Fx(\()p Fv(M)10 b Fx(\)\()1894 2041 y Fr(r)1928 2018 y Fh(0)p 1894 2058 57 4 v 1895 2115 a Fr(s)1928 2096 y Fh(0)1970 2041 y Fr(r)p 1970 2058 34 4 v 1971 2115 a(s)2043 2081 y Fq(\012)29 b Fv(m)p Fx(\))44 b(=)h Fv(\013)q Fx(\()p Fv(M)10 b Fx(\)\()2728 2041 y Fr(r)2762 2018 y Fh(0)2784 2041 y Fr(r)p 2728 2058 90 4 v 2729 2115 a(s)2762 2096 y Fh(0)2784 2115 y Fr(s)2857 2081 y Fq(\012)29 b Fv(m)p Fx(\))45 b(=)3261 2041 y Fr(r)3295 2018 y Fh(0)3317 2041 y Fr(r)r(m)p 3261 2058 153 4 v 3293 2115 a(s)3326 2096 y Fh(0)3348 2115 y Fr(s)3467 2081 y Fx(=)3598 2041 y Fr(r)3632 2018 y Fh(0)p 3598 2058 57 4 v 3599 2115 a Fr(s)3632 2096 y Fh(0)3674 2041 y Fr(r)r(m)p 3674 2058 97 4 v 3705 2115 a(s)3824 2081 y Fx(=)10 2174 y Fr(r)44 2151 y Fh(0)p 10 2191 57 4 v 11 2248 a Fr(s)44 2229 y Fh(0)76 2214 y Fv(\013)q Fx(\()p Fv(M)10 b Fx(\)\()367 2174 y Fr(r)p 367 2191 34 4 v 368 2248 a(s)433 2214 y Fq(\012)23 b Fv(m)p Fx(\).)0 2330 y Fv(\013)33 b Fx(is)g(a)g (functorial)f(homomorphism.)45 b(In)33 b(fact)f(the)h(diagram)1338 3049 y Fv(S)1404 3013 y Fp(\000)p Fk(1)1498 3049 y Fv(R)23 b Fq(\012)1672 3064 y Fr(R)1753 3049 y Fv(N)609 b(S)2506 3013 y Fp(\000)p Fk(1)2600 3049 y Fv(N)p 1871 3019 542 4 v 2330 3017 a Fj(-)2027 3111 y Fv(\013)q Fx(\()p Fv(N)10 b Fx(\))1330 2562 y Fv(S)1396 2525 y Fp(\000)p Fk(1)1490 2562 y Fv(R)23 b Fq(\012)1664 2577 y Fr(R)1745 2562 y Fv(M)593 b(S)2498 2525 y Fp(\000)p Fk(1)2592 2562 y Fv(M)p 1878 2531 526 4 v 2321 2529 a Fj(-)2019 2485 y Fv(\013)q Fx(\()p Fv(M)10 b Fx(\))p 1588 2949 4 351 v 1590 2949 a Fj(?)1077 2803 y Fv(S)1143 2767 y Fp(\000)p Fk(1)1237 2803 y Fv(R)24 b Fq(\012)1412 2818 y Fr(R)1492 2803 y Fv(f)p 2563 2949 V 2565 2949 a Fj(?)2604 2803 y Fv(S)2670 2767 y Fp(\000)p Fk(1)2764 2803 y Fv(f)0 3295 y Fx(comm)m(utes)42 b(since)g(w)m(e)f(ha)m(v)m(e)h Fv(S)1163 3259 y Fp(\000)p Fk(1)1257 3295 y Fv(f)c Fq(\016)27 b Fv(\013)q Fx(\()p Fv(M)10 b Fx(\)\()1711 3256 y Fr(r)p 1711 3272 34 4 v 1712 3329 a(s)1783 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b(o)g(of.)41 b Fx(This)34 b(follo)m(ws)g(from)e(Corollary)h(6.13,)g (Exercise)i(5.2)d(\(1\))h(and)g(Exercise)i(6.2.)44 b(W)-8 b(e)33 b(giv)m(e)h(a)e(direct)0 5490 y(pro)s(of.)43 b(Let)1466 5646 y Fv(M)1560 5661 y Fk(1)1666 5586 y Fr(f)1628 5646 y Fq(\000)-60 b(!)28 b Fv(M)1867 5661 y Fk(2)1975 5586 y Fr(g)1934 5646 y Fq(\000)-60 b(!)28 b Fv(M)2173 5661 y Fk(3)2240 5646 y Fq(\000)-59 b(!)27 b Fx(0)p eop end %%Page: 84 84 TeXDict begin 84 83 bop 0 -170 a Fu(84)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fx(b)s(e)32 b(exact.)44 b(This)33 b(is)g(equiv)-5 b(alen)m(t)34 b(to)d Fv(g)k Fx(surjectiv)m(e,)g Fv(g)t(f)i Fx(=)28 b(0)j(and)i(Ke\()p Fv(g)t Fx(\))27 b Fq(\022)h Fx(Im\()p Fv(f)11 b Fx(\).)43 b(The)33 b(map)f Fv(P)i Fq(\012)3673 44 y Fr(R)3752 29 y Fv(g)h Fx(is)0 146 y(surjectiv)m(e,)k(for)621 71 y Fl(P)742 146 y Fv(p)791 161 y Fr(i)844 146 y Fq(\012)25 b Fv(m)1031 161 y Fr(i)p Fk(3)1128 146 y Fx(=)1238 71 y Fl(P)1360 146 y Fv(p)1409 161 y Fr(i)1461 146 y Fq(\012)g Fv(g)t 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Fq(\012)g Fv(A)2956 3161 y Fr(n)3003 3146 y Fx(.)0 3262 y(The)35 b(coherence)h(theorem)e(of)g(S.)g(MacLane)h(sa)m(ys)g(that)f(all)g (diagrams)g(whose)h(morphisms)h(are)e(formed)0 3378 y(using)i Fv(\013)q Fx(,)f Fv(\025)p Fx(,)h Fv(\032)p Fx(,)g(iden)m(tities,)i(in) m(v)m(erses,)h(tensor)d(pro)s(ducts,)h(and)e(comp)s(ositions)h(thereof) g(comm)m(ute.)53 b(W)-8 b(e)0 3495 y(will)36 b(not)g(pro)m(v)m(e)h (this)f(theorem.)54 b(It)36 b(implies)h(that)f(eac)m(h)g(monoidal)g (category)g(can)g(b)s(e)g(replaced)h(b)m(y)f(\(is)0 3611 y(monoidally)j(equiv)-5 b(alen)m(t)40 b(to\))e(a)g(strict)h(monoidal)f (category)-8 b(,)40 b(that)e(is)h(in)f(all)g(diagrams)h(w)m(e)g(ma)m(y) g(omit)0 3727 y(the)33 b(morphisms)h Fv(\013)q(;)17 b(\025;)g(\032)p Fx(,)32 b(i.)h(e.)44 b(replace)33 b(them)g(b)m(y)h(iden)m(tities.)45 b(In)33 b(particular)g(on)f Fv(A)3133 3742 y Fk(1)3195 3727 y Fq(\012)23 b Fv(:)17 b(:)g(:)k Fq(\012)i Fv(A)3604 3742 y Fr(n)3683 3727 y Fx(there)0 3843 y(is)33 b(only)g(one)g (automorphism)g(formed)g(with)h(coherence)g(morphisms,)g(the)f(iden)m (tit)m(y)-8 b(.)0 4012 y Ft(Remark)54 b(11.3.)49 b Fx(F)-8 b(or)46 b(eac)m(h)i(monoidal)f(category)g Fq(C)53 b Fx(one)47 b(can)g(construct)h(the)f(monoidal)g(category)0 4129 y Fq(C)58 4092 y Fr(sy)r(mm)294 4129 y Fx(symmetric)38 b(to)f Fq(C)42 b Fx(whic)m(h)c(coincides)h(with)e Fq(C)43 b Fx(as)36 b(a)h(category)-8 b(,)38 b(whic)m(h)g(has)f(the)g(tensor)g (pro)s(duct)0 4245 y Fv(A)22 b Fo(\002)h Fv(B)33 b Fx(:=)27 b Fv(B)h Fq(\012)22 b Fv(A)p Fx(,)33 b(and)g(coherence)h(morphisms)988 4394 y Fv(\013)q Fx(\()p Fv(C)r(;)17 b(B)5 b(;)17 b(A)p Fx(\))1439 4358 y Fp(\000)p Fk(1)1560 4394 y Fx(:)28 b(\()p Fv(A)22 b Fo(\002)h Fv(B)5 b Fx(\))22 b Fo(\002)h Fv(C)34 b Fq(\000)-59 b(!)27 b Fv(A)22 b Fo(\002)h Fx(\()p Fv(B)k Fo(\002)c Fv(C)7 b Fx(\))p Fv(;)988 4511 y(\032)p Fx(\()p Fv(A)p Fx(\))28 b(:)g Fv(I)i Fo(\002)23 b Fv(A)28 b Fq(\000)-60 b(!)27 b Fv(A;)988 4627 y(\025)p Fx(\()p Fv(A)p Fx(\))h(:)g Fv(A)22 b Fo(\002)h Fv(I)35 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Fq(\010)1192 5429 y Fr(g)r Fp(2)p Fr(G)1335 5414 y Fv(V)1392 5429 y Fr(g)1464 5414 y Fx(holds.)0 5530 y(Let)h Fv(V)48 b Fx(and)28 b Fv(W)41 b Fx(b)s(e)27 b Fv(G)p Fx(-graded)g(v)m(ector)i(spaces.)43 b(A)27 b(linear)h(map)g Fv(f)38 b Fx(:)28 b Fv(V)49 b Fq(\000)-59 b(!)27 b Fv(W)41 b Fx(is)28 b(called)g Fs(of)i(de)-5 b(gr)g(e)g(e)29 b Fv(e)f Fq(2)g Fv(G)p Fx(,)0 5646 y(if)k(for)g(all)h Fv(g)e Fq(2)d Fv(G)k(f)11 b Fx(\()p Fv(V)809 5661 y Fr(g)849 5646 y Fx(\))28 b Fq(\022)g Fv(W)1112 5661 y Fr(g)1185 5646 y Fx(holds.)p eop end %%Page: 88 88 TeXDict begin 88 87 bop 0 -170 a Fu(88)1389 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)0 29 y Fx(The)44 b Fv(G)p Fx(-graded)g(v)m(ector)g(spaces)h(and)f(linear)g(maps)g(of)f(degree)h Fv(e)j Fq(2)g Fv(G)c Fx(form)g(the)h(category)g Fq(M)3719 -7 y Fr(G)3821 29 y Fx(of)0 146 y Fv(G)p Fs(-gr)-5 b(ade)g(d)34 b(ve)-5 b(ctor)35 b(sp)-5 b(ac)g(es)p Fx(.)0 262 y Fq(M)120 226 y Fr(G)227 262 y Fx(carries)49 b(a)e(monoidal)i(structure)g(with)f 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b Fq(\012)h Fv(A)701 b(B)27 b Fq(\012)c Fv(B)p 1624 4386 644 4 v 2185 4384 a Fj(-)1877 4352 y Fr(f)7 b Fp(\012)p Fr(f)p 1459 4803 4 351 v 1461 4803 a Fj(?)1313 4645 y Fp(r)1372 4656 y Fi(A)p 2434 4803 V 2436 4803 a Fj(?)2475 4645 y Fp(r)2534 4656 y Fi(B)1425 4905 y Fv(A)898 b(B)p 1526 4873 842 4 v 2285 4871 a Fj(-)1926 4937 y Fr(f)0 5076 y Fx(and)1923 5160 y Fv(I)1668 5647 y(A)1682 5377 y Fr(\021)1717 5388 y Fi(A)1873 5277 y Fj(\001)1831 5360 y(\001)1790 5443 y(\001)1748 5526 y(\001)1739 5545 y(\001)-42 b(\013)2153 5647 y Fv(B)2130 5377 y Fr(\021)2165 5388 y Fi(B)1983 5277 y Fj(A)2024 5360 y(A)2066 5443 y(A)2107 5526 y(A)2117 5545 y(A)g(U)p 1734 5615 429 4 v 2080 5613 a(-)1928 5581 y Fr(f)p eop end %%Page: 91 91 TeXDict begin 91 90 bop 1605 -170 a Fu(Monoidal)27 b(Categories)1531 b(91)0 29 y Ft(Remark)28 b(11.8.)35 b Fx(Ob)m(viously)26 b(the)e(comp)s(osition)h(of)e(t)m(w)m(o)i(morphisms)h(of)d(algebras)h (is)h(again)e(a)h(morphism)0 146 y(of)42 b(algebras.)72 b(Also)43 b(the)g(iden)m(tit)m(y)h(morphism)f(is)g(a)f(morphism)h(of)f (algebras.)72 b(Th)m(us)44 b(w)m(e)f(obtain)g(the)0 262 y(category)33 b(Alg\()p Fq(C)6 b Fx(\))33 b(of)f(algebras)h(in)f Fq(C)6 b Fx(.)0 436 y Ft(De\014nition)28 b(11.9.)35 b Fx(A)24 b Fs(c)-5 b(o)g(algebr)g(a)22 b Fx(or)h(a)h Fs(c)-5 b(omonoid)22 b Fx(in)i(a)f(monoidal)h(category)g Fq(C)30 b Fx(is)24 b(an)g(ob)5 b(ject)24 b Fv(C)31 b Fx(together)0 552 y(with)i(a)f(com)m(ultiplication)j(\001)28 b(:)f Fv(C)35 b Fq(\000)-60 b(!)28 b Fv(C)h Fq(\012)22 b Fv(C)7 b Fx(,)33 b(that)f(is)h(coasso)s(ciativ)m(e:)1374 715 y Fv(C)806 b(C)29 b Fq(\012)23 b Fv(C)p 1480 687 741 4 v 2138 685 a Fj(-)1821 666 y Fk(\001)p 1411 1104 4 351 v 1413 1104 a Fj(?)1315 952 y Fk(\001)p 2386 1104 V 2388 1104 a Fj(?)2427 949 y Fk(\001)p Fp(\012)p Fk(id)1275 1203 y Fv(C)29 b Fq(\012)23 b Fv(C)607 b(C)29 b Fq(\012)22 b Fv(C)29 b Fq(\012)23 b Fv(C)p 1580 1175 543 4 v 2040 1173 a Fj(-)1758 1238 y Fk(id)12 b Fp(\012)p Fk(\001)0 1363 y Fx(and)33 b(a)f(counit)h 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Ft(Problem)47 b(12.2.)f Fx(\(1\))41 b(Let)g Fv(B)46 b Fx(b)s(e)41 b(a)f(bialgebra)h(and)g Fq(M)2187 3325 y Fr(B)2288 3310 y Fx(b)s(e)g(the)h(category)f(of)f (righ)m(t)h Fv(B)5 b Fx(-)41 b(mo)s(dules.)0 3427 y(Sho)m(w)33 b(that)g Fq(M)589 3442 y Fr(B)682 3427 y Fx(is)g(a)f(monoidal)h (category)-8 b(.)0 3543 y(\(2\))33 b(Let)h Fv(B)39 b Fx(a)34 b(bialgebra)g(and)f Fq(M)1264 3507 y Fr(B)1359 3543 y Fx(b)s(e)g(the)i(category)f(of)f(righ)m(t)h Fv(B)5 b Fx(-)33 b(como)s(dules.)49 b(Sho)m(w)34 b(that)g Fq(M)3658 3507 y Fr(B)3752 3543 y Fx(is)g(a)0 3659 y(monoidal)f(category)-8 b(.)0 3839 y Ft(De\014nition)51 b(12.3.)c Fx(\(1\))c(Let)h(\()p Fv(B)5 b(;)17 b Fq(r)p Fv(;)g(\021)t(;)g Fx(\001)p Fv(;)g(\017)p Fx(\))42 b(b)s(e)i(a)f(bialgebra.)76 b(Let)44 b Fv(A)f Fx(b)s(e)h(a)f(left)h Fv(B)5 b Fx(-mo)s(dule)43 b(with)0 3955 y(structure)30 b(map)e Fv(\026)g Fx(:)f Fv(B)19 b Fq(\012)14 b Fv(A)28 b Fq(\000)-60 b(!)28 b Fv(A)p Fx(.)42 b(Let)28 b(furthermore)h(\()p Fv(A;)17 b Fq(r)2282 3970 y Fr(A)2339 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Fx(=)f(0.)347 4609 y(\(2\))41 b(F)s(\177)-51 b(ur)32 b(jeden)h Fv(R)q Fx(-Mo)s(dul)f(gilt) h Fv(R)23 b Fq(\012)1723 4624 y Fr(R)1803 4609 y Fv(M)1936 4582 y Fq(\030)1937 4613 y Fx(=)2041 4609 y Fv(M)10 b Fx(.)347 4747 y(\(3\))41 b(Sei)33 b(der)g Fn(Q)p Fx(-V)-8 b(ektorraum)34 b Fv(V)49 b Fx(=)27 b Fn(Q)1774 4711 y Fr(n)1854 4747 y Fx(gegeb)s(en.)597 4863 y(\(a\))41 b(Bestimme)35 b(dim)1380 4878 y Fm(R)1432 4863 y Fx(\()p Fn(R)23 b Fq(\012)1642 4878 y Fm(Q)1720 4863 y Fv(V)f Fx(\).)592 5001 y(\(b\))41 b(Gib)33 b(explizit)h(einen)g(Isomorphism)m(us)h Fn(R)22 b Fq(\012)2392 5016 y Fm(Q)2470 5001 y Fv(V)2577 4973 y Fq(\030)2577 5005 y Fx(=)2682 5001 y Fn(R)2754 4965 y Fr(n)2833 5001 y Fx(an.)347 5139 y(\(4\))41 b(Sei)33 b Fv(V)54 b Fx(ein)34 b Fn(Q)p Fx(-V)-8 b(ektorraum)33 b(und)g Fv(W)46 b Fx(ein)33 b Fn(R)p Fx(-V)-8 b(ektorraum.)597 5255 y(\(a\))41 b(Hom)967 5270 y Fm(R)1019 5255 y Fx(\()p Fv(:)p Fn(R)1156 5270 y Fm(Q)1212 5255 y Fv(;)17 b(:W)d Fx(\))1454 5227 y Fq(\030)1454 5259 y Fx(=)1559 5255 y Fv(W)46 b Fx(in)33 b Fn(Q)p Fx(-Mo)s(d.)592 5393 y(\(b\))41 b(Hom)967 5408 y Fm(Q)1022 5393 y Fx(\()p Fv(:V)5 b(;)17 b(:W)d Fx(\))1392 5365 y Fq(\030)1393 5397 y Fx(=)1497 5393 y(Hom)1700 5408 y Fm(R)1752 5393 y Fx(\()p Fv(:)p Fn(R)22 b Fq(\012)1988 5408 y Fm(Q)2067 5393 y Fv(V)5 b(;)17 b(:W)d Fx(\).)603 5530 y(\(c\))41 b(Sei)d(dim)1087 5545 y Fm(Q)1143 5530 y Fv(V)57 b(<)35 b Fq(1)i Fx(und)g(dim)1866 5545 y Fm(R)1919 5530 y Fv(W)49 b(<)35 b Fq(1)p Fx(.)56 b(Wie)38 b(k)-5 b(ann)37 b(man)g(v)m(erstehen,)k(da\031)c(in)g(4b)763 5646 y(links)d(unendlic)m(he)h(Matrizen)f(und)f(rec)m(h)m(ts)h(endlic)m (he)h(Matrizen)f(stehen?)p eop end %%Page: 102 102 TeXDict begin 102 101 bop 0 -170 a Fu(102)1351 b(Adv)l(anced)24 b(Algebra)i({)g(P)n(areigis)592 29 y Fx(\(d\))41 b(Hom)967 44 y Fm(Q)1022 29 y Fx(\()p Fv(:V)5 b(;)17 b Fx(Hom)1397 44 y Fm(R)1449 29 y Fx(\()p Fv(:)p Fn(R)p Fv(;)g(:W)d Fx(\))1828 2 y Fq(\030)1829 34 y Fx(=)1933 29 y(Hom)2136 44 y Fm(R)2188 29 y Fx(\()p Fv(:)p Fn(R)22 b Fq(\012)2424 44 y Fm(Q)2503 29 y Fv(V)5 b(;)17 b(:W)d Fx(\).)347 167 y(\(5\))41 b Fn(Z)p Fv(=)p Fx(\(18\))22 b Fq(\012)901 182 y Fm(Z)972 167 y Fn(Z)p Fv(=)p Fx(\(30\))27 b Fq(6)p Fx(=)h(0.)347 305 y(\(6\))41 b Fv(m)28 b Fx(:)g Fn(Z)p Fv(=)p Fx(\(18\))16 b Fq(\012)1063 320 y Fm(Z)1128 305 y Fn(Z)p Fv(=)p Fx(\(30\))27 b Fq(3)p 1538 250 56 4 v 28 w Fv(x)17 b Fq(\012)p 1703 250 52 4 v 16 w Fv(y)31 b Fq(7!)p 1910 250 107 4 v 28 w Fv(xy)g Fq(2)d Fn(Z)p Fv(=)p Fx(\(6\))h(ist)h(ein)h(Homomorphism)m(us)g(und)g Fv(m)e Fx(ist)513 421 y(bijektiv.)347 559 y(\(7\))41 b(F)s(\177)-51 b(ur)32 b Fn(Q)p Fx(-V)-8 b(ektorr\177)-49 b(aume)33 b Fv(V)54 b Fx(und)33 b Fv(W)46 b Fx(gilt)33 b Fv(V)44 b Fq(\012)2190 574 y Fm(Z)2261 559 y Fv(W)2394 531 y Fq(\030)2395 563 y Fx(=)2499 559 y Fv(V)g Fq(\012)2677 574 y Fm(Q)2755 559 y Fv(W)14 b Fx(.)347 696 y(\(8\))41 b(F)s(\177)-51 b(ur)32 b(jede)h(endlic)m(he)h(ab)s(elsc)m(he)h(Grupp)s (e)d Fv(M)44 b Fx(gilt)32 b Fn(Q)23 b Fq(\012)2510 711 y Fm(Z)2581 696 y Fv(M)38 b Fx(=)28 b(0.)347 834 y(\(9\))41 b Fn(Z)p Fv(=)p Fx(\()p Fv(m)p Fx(\))23 b Fq(\012)889 849 y Fm(Z)960 834 y Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))1237 806 y Fq(\030)1238 838 y Fx(=)1342 834 y Fn(Z)p Fv(=)p Fx(\(ggT\()p Fv(m;)17 b(n)p Fx(\)\).)298 972 y(\(10\))41 b Fn(Q)23 b Fq(\012)690 987 y Fm(Z)761 972 y Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))28 b(=)f(0.)298 1109 y(\(11\))41 b(Hom)716 1124 y Fm(Z)765 1109 y Fx(\()p Fn(Q)p Fv(;)17 b Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\)\))28 b(=)f(0.)298 1247 y(\(12\))41 b(Gib)33 b(explizit)h(Isomorphismen)g(an)f(f)s(\177) -51 b(ur)1650 1403 y Fn(Z)22 b Fq(\012)1815 1418 y Fm(Z)1886 1403 y Fn(Q)1992 1375 y Fq(\030)1992 1407 y Fx(=)2097 1403 y Fn(Q)p Fv(;)1650 1519 y Fx(3)p Fn(Z)g Fq(\012)1864 1534 y Fm(Z)1935 1519 y Fn(Q)2040 1491 y Fq(\030)2041 1523 y Fx(=)2146 1519 y Fn(Q)p Fv(:)513 1672 y Fx(Zeige,)33 b(da\031)f(das)h(Diagramm)f(k)m(omm)m(utiert:)1680 2297 y Fn(Q)410 b(Q)p 1786 2273 353 4 v 2056 2271 a Fj(-)1924 2353 y Fx(3)p Fq(\001)1537 1809 y Fx(3)p Fn(Z)22 b Fq(\012)1751 1824 y Fm(Z)1822 1809 y Fn(Q)150 b(Z)22 b Fq(\012)2214 1824 y Fm(Z)2285 1809 y Fn(Q)p 1929 1785 91 4 v 1937 1783 a Fj(-)p 1717 2202 4 351 v 1718 2202 a(?)1602 2029 y Fq(\030)1603 2061 y Fx(=)p 2204 2202 V 2206 2202 a Fj(?)2245 2029 y Fq(\030)2246 2061 y Fx(=)298 2477 y(\(13\))41 b(Der)27 b(Homomorphism)m(us)i(2)p Fn(Z)11 b Fq(\012)1691 2492 y Fr(Z)1759 2477 y Fn(Z)p Fv(=)p Fx(\(2\))28 b Fq(\000)-60 b(!)27 b Fn(Z)11 b Fq(\012)2325 2492 y Fm(Z)2385 2477 y Fn(Z)p Fv(=)p Fx(\(2\))27 b(ist)h(der)f(Nullhomomorphism)m(us,)513 2593 y(b)s(eide)34 b(Mo)s(duln)f(sind)g(ab)s(er)g(v)m(on)g(Null)g(v)m (ersc)m(hieden.)135 2752 y(I)s(I)s(I.)42 b(Pro)5 b(jektiv)m(e)34 b(Mo)s(duln)347 2889 y(\(1\))41 b(Bestimme)35 b(die)e(Dual-Basis)f(v)m (on)i Fn(R)1873 2853 y Fk(2)1944 2889 y Fx(im)f(Sinne)h(der)f(V)-8 b(orlesung.)347 3027 y(\(2\))41 b(Zeige,)33 b(da\031)f(die)h(Spur)h (eines)g(Homomorphism)m(us)g Fv(f)39 b Fx(:)27 b Fv(V)50 b Fq(\000)-60 b(!)27 b Fv(V)54 b Fx(gegeb)s(en)34 b(ist)f(durc)m(h)1384 3196 y(End)1558 3211 y Fr(K)1627 3196 y Fx(\()p Fv(V)21 b Fx(\))1809 3168 y Fq(\030)1810 3200 y Fx(=)1914 3196 y Fv(V)44 b Fq(\012)22 b Fv(V)2193 3155 y Fp(\003)2284 3139 y Fk(ev)2260 3196 y Fq(\000)-60 b(!)28 b Fv(K)r(:)347 3353 y Fx(\(3\))41 b(Bestimme)35 b(die)e(Dual-Basis)f(v)m(on)1801 3368 y Fr(R)p Fp(\002)p Fr(S)1960 3353 y Fv(R)23 b Fq(\002)g Fx(0)k Fq(\022)h Fv(R)23 b Fq(\002)g Fv(S)6 b Fx(.)347 3491 y(\(4\))41 b Fv(K)596 3506 y Fr(n)676 3491 y Fx(ist)33 b(ein)g(pro)5 b(jektiv)m(er)34 b Fv(M)1561 3506 y Fr(n)1609 3491 y Fx(\()p Fv(K)7 b Fx(\)-Mo)s(dul.)347 3629 y(\(5\))41 b(Sei)33 b Fv(R)c Fx(:=)f Fv(K)h Fq(\002)23 b Fv(K)39 b Fx(mit)33 b(einem)h(K\177)-49 b(orp)s(er)32 b Fv(K)7 b Fx(.)597 3745 y(\(a\))41 b(Zeige:)j Fv(P)d Fx(:=)28 b Fq(f)p Fx(\()p Fv(a;)17 b Fx(0\))p Fq(j)p Fv(a)27 b Fq(2)h Fv(K)7 b Fq(g)32 b Fx(ist)h(ein)g(endlic)m(h)i(erzeugter)e(pro)5 b(jektiv)m(er)35 b Fv(R)q Fx(-Mo)s(dul.)592 3883 y(\(b\))41 b(Sind)34 b(die)f Fv(R)q Fx(-Mo)s(duln)f Fv(P)46 b Fx(und)33 b Fv(Q)28 b Fx(:=)g Fq(f)p Fx(\(0)p Fv(;)17 b(a)p Fx(\))p Fq(j)p Fv(a)27 b Fq(2)h Fv(K)7 b Fq(g)33 b Fx(isomorph?)603 4020 y(\(c\))41 b(Man)33 b(\014nde)h(eine)f(Dual-Basis)g(f)s(\177)-51 b(ur)31 b Fv(P)14 b Fx(.)347 4158 y(\(6\))41 b(Zeige)29 b(f)s(\177)-51 b(ur)29 b Fv(R)f Fx(:=)g Fv(M)1243 4173 y Fr(n)1290 4158 y Fx(\()p Fv(K)7 b Fx(\),)30 b(da\031)f Fv(P)41 b Fx(=)28 b Fv(K)1985 4173 y Fr(n)2061 4158 y Fx(endlic)m(h)i(erzeugt)g(pro)5 b(jektiv)31 b(ist,)f(und)g(\014nde)g (eine)513 4274 y(Dual-Basis.)347 4412 y(\(7\))41 b(Zu)33 b(jedem)g(pro)5 b(jektiv)m(en)35 b(Mo)s(dul)e Fv(P)46 b Fx(gibt)32 b(es)i(einen)f(freien)h(Mo)s(dul)e Fv(F)47 b Fx(mit)33 b Fv(P)i Fq(\010)23 b Fv(F)3614 4384 y Fq(\030)3615 4416 y Fx(=)3719 4412 y Fv(F)14 b Fx(.)138 4571 y(IV.)42 b(Kategorien)32 b(und)h(F)-8 b(unktoren)347 4708 y(\(1\))41 b(In)33 b Fv(R)q Fx(-Mo)s(d)f(gilt:)513 4824 y Fv(f)39 b Fx(:)27 b Fv(M)39 b Fq(\000)-60 b(!)28 b Fv(N)43 b Fx(Monomorphism)m(us)62 b Fq(\()-17 b(\))60 b Fv(f)43 b Fx(injektiv)m(er)35 b(Homomorphism)m(us.)347 4962 y(\(2\))125 b(\(a\))41 b(W)-8 b(enn)48 b Fv(f)64 b Fx(:)52 b Fv(M)64 b Fq(\000)-60 b(!)52 b Fv(N)58 b Fx(surjektiv)49 b(ist,)i(dann)d(ist)g (Hom)2921 4977 y Fr(R)2978 4962 y Fx(\()p Fv(f)5 b(;)17 b(P)d Fx(\))52 b(:)h(Hom)3563 4977 y Fr(R)3621 4962 y Fx(\()p Fv(N)5 b(;)17 b(P)d Fx(\))763 5078 y Fq(\000)-59 b(!)27 b Fx(Hom)1111 5093 y Fr(R)1169 5078 y Fx(\()p Fv(M)5 b(;)17 b(P)d Fx(\))32 b(injektiv.)592 5216 y(\(b\))41 b Fn(Z)28 b Fq(\000)-59 b(!)27 b Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))33 b(induziert)h(eine)f(injektiv)m(e)i(Abbildung)1109 5373 y(Hom)1312 5388 y Fm(Z)1360 5373 y Fx(\()p Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))p Fv(;)17 b(M)10 b Fx(\))29 b Fq(\000)-60 b(!)27 b Fx(Hom)2209 5388 y Fm(Z)2258 5373 y Fx(\()p Fn(Z)p Fv(:M)10 b Fx(\))2560 5345 y Fq(\030)2561 5377 y Fx(=)2665 5373 y Fv(M)5 b(:)763 5530 y Fx(W)-8 b(arum)43 b(k)-5 b(ann)42 b(man)g(Hom)1794 5545 y Fm(Z)1843 5530 y Fx(\()p Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))p Fv(;)17 b(M)10 b Fx(\))42 b(mit)h Fq(f)p Fv(x)h Fq(2)g Fv(M)10 b Fq(j)p Fv(nx)45 b Fx(=)f(0)p Fq(g)f(\022)i Fv(M)52 b Fx(iden)m(ti-)763 5646 y(\014zieren?)p eop end %%Page: 103 103 TeXDict begin 103 102 bop 1443 -170 a Fu(Quic)n(kies)26 b(in)f(Adv)l(anced)f(Algebra)1329 b(103)603 29 y Fx(\(c\))41 b Fv(T)820 44 y Fr(n)868 29 y Fx(\()p Fv(M)10 b Fx(\))28 b(:=)g Fq(f)p Fv(x)f Fq(2)i Fv(M)10 b Fq(j)p Fv(nx)28 b Fx(=)g(0)p Fq(g)k Fx(ist)h(ein)g(F)-8 b(unktor)33 b(Ab)28 b Fq(\000)-60 b(!)27 b Fx(Ab)q(.)592 167 y(\(d\))41 b(Die)33 b(Ein)m(b)s(ettung)h Fv(T)1509 182 y Fr(n)1556 167 y Fx(\()p Fv(M)10 b Fx(\))28 b Fq(\000)-59 b(!)27 b Fv(M)43 b Fx(ist)33 b(ein)g(funktorieller)h(Homomorphism)m(us.)347 305 y(\(3\))41 b(In)33 b Fv(R)q Fx(-Mo)s(d)f(gilt:)513 421 y Fv(f)39 b Fx(:)27 b Fv(M)39 b Fq(\000)-60 b(!)28 b Fv(N)43 b Fx(Epimorphism)m(us)63 b Fq(\()-17 b(\))60 b Fv(f)43 b Fx(surjektiv.)347 559 y(\(4\))e(W)-8 b(enn)26 b Fq(F)34 b Fx(ein)25 b(k)m(o)m(v)-5 b(arian)m(ter)26 b(darstellbarer)g(F)-8 b(unktor)24 b(ist)h(und)h Fv(f)38 b Fx(:)28 b Fv(M)38 b Fq(\000)-59 b(!)27 b Fv(N)35 b Fx(ein)25 b(Monomor-)513 675 y(phism)m(us)35 b(ist,)e(dann)g(ist)g Fq(F)10 b Fx(\()p Fv(f)h Fx(\))27 b(:)h Fq(F)10 b Fx(\()p Fv(M)g Fx(\))27 b Fq(\000)-59 b(!)27 b(F)10 b Fx(\()p Fv(N)g Fx(\))33 b(eb)s(enfalls)g(ein)g(Monomorphism)m(us.)347 813 y(\(5\))41 b(Der)33 b(F)-8 b(unktor)32 b Fq(F)37 b Fx(:)28 b Fv(M)39 b Fq(7!)27 b Fn(Z)p Fv(=)p Fx(\()p Fv(n)p Fx(\))22 b Fq(\012)1846 828 y Fm(Z)1917 813 y Fv(M)44 b Fx(ist)33 b(nic)m(h)m(t)g(darstellbar.)347 950 y(\(6\))41 b(Der)33 b(F)-8 b(unktor)32 b Fq(F)37 b Fx(:)28 b Fv(V)49 b Fq(7!)28 b Fn(Q)1549 914 y Fr(n)1618 950 y Fq(\012)1695 965 y Fm(Q)1774 950 y Fv(V)54 b Fx(ist)33 b(darstellbar.)347 1088 y(\(7\))41 b(Der)33 b(F)-8 b(unktor)32 b Fv(T)1131 1103 y Fr(n)1206 1088 y Fx(:)c(Ab)g Fq(\000)-60 b(!)27 b Fx(Ab)33 b(mit)g Fv(T)1956 1103 y Fr(n)2003 1088 y Fx(\()p Fv(M)10 b Fx(\))29 b(:=)e Fq(f)p Fv(x)h Fq(2)g Fv(M)10 b Fq(j)p Fv(nx)29 b Fx(=)f(0)p Fq(g)k Fx(ist)h(darstellbar.)347 1226 y(\(8\))41 b(Jeder)36 b(additiv)m(e)h(F)-8 b(unktor)35 b Fv(F)46 b Fx(:)32 b Fv(R)q Fx(-Mo)s(d)g Fq(\000)-59 b(!)32 b Fv(S)6 b Fx(-)o(Mo)s(d)35 b(erh\177)-49 b(alt)36 b(endlic)m(he)h(direkte)f(Summen,)513 1342 y(d.h.)44 b Fv(F)14 b Fx(\()p Fv(M)33 b Fq(\010)22 b Fv(N)10 b Fx(\))1214 1314 y Fq(\030)1215 1346 y Fx(=)1320 1342 y Fv(F)k Fx(\()p Fv(M)c Fx(\))22 b Fq(\010)h Fv(F)14 b Fx(\()p Fv(N)c Fx(\).)173 1509 y(V.)42 b(Morita-)650 1484 y(\177)638 1509 y(Aquiv)-5 b(alenz)347 1655 y(\(1\))41 b(Zeige,)33 b(da\031)f(\()p Fv(K)e Fq(\002)22 b Fv(K)7 b Fx(\)-Mo)s(d)32 b(nic)m(h)m(t)i(\177)-49 b(aquiv)-5 b(alen)m(t)34 b(zu)f Fv(K)7 b Fx(-Mo)s(d)32 b(ist.)347 1793 y(\(2\))41 b(Sei)28 b Fv(K)33 b Fx(ein)28 b(K\177)-49 b(orp)s(er,)27 b Fv(B)33 b Fx(:=)28 b Fv(M)1620 1808 y Fr(n)1667 1793 y Fx(\()p Fv(K)7 b Fx(\),)1888 1808 y Fr(K)1956 1793 y Fv(P)2019 1808 y Fr(B)2108 1793 y Fx(:=)27 b Fv(K)2328 1757 y Fr(n)2402 1793 y 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(cyclic)i(group)e(with)h(t)m(w)m(o)g(elemen)m(ts.,)i Fv(A)28 b Fx(:=)g Fn(C)p Fx(,)i(and)h Fn(K)d Fx(:=)f Fn(R)p Fx(.)43 b(Sho)m(w)513 3241 y(that)33 b Fv(')27 b Fx(:)h Fn(C)p Fx(#)p Fn(R)p Fv(C)1166 3256 y Fk(2)1233 3241 y Fq(\000)-60 b(!)28 b Fx(End)1553 3256 y Fm(R)1605 3241 y Fx(\()p Fn(C)p Fx(\))k(is)h(an)g(isomorphism)h(of)e(algebras.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF