; TeX output 2001.10.26:11387 V6N cmbx12MathematischesInstitutmWS2001/026derUniversitXatM`unchenSet16Prof.Dr.B.Pareigis;KNff cmbx12ProblemffsetforAdvancedffAlgebra=}HXQ cmr12(1)\$Let`g cmmi12RbSearingandM$Danabeliangroup.6sShorwthatthereisa\$one-to-oneLcorrespSondencebetrweenLmapsfQ:URR!", cmsy10gM6!Mkthat\$makre0YMq=intoaleftRJ-moSduleandringhomomorphisms(always\$preservingtheunitelemenrt)gË:URRn'!{End):(M@).H(2)\$LetfQ:URM6!N禹bSeanRJ-modulehomomorphism.Thefollorwing\$areequivXalenrt:`(a)tPf2isamonomorphism,`(b)tPforallRJ-moSdulesPwandallhomomorphismsgn9;hUR:P!eMefGgË=URfh=)gË=h;aU(c)tPforallRJ-moSdulesPnthehomomorphismofabeliangroupszHomi2cmmi8R(PS;fG)UR:?HomR&(P;M@)UR3gË7!fGg2Hom۟R"n(PS;N@)tPisamonomorphism.H(3)\$AreŨf(0;:::ʜ;a;:::;0)ja2KnPgŨandf(a;0;:::;0)ja2KnPgŨiso-\$morphicasMnP(Kܞ)-moSdules?H(4)\$Shorw:mw:( msbm10Z=(18) )ppmsbm8Z Z=(30)3dzRKx dz(ߟKy{7!dz 1Kxy2Z=(6)/isa\$homomorphismandmisbijectivre.6Duedate:8TVuesdary,23.10.2001,16:15inLectureHallE06*;7 )ppmsbm8( msbm10!", cmsy102cmmi8g cmmi12Nff cmbx12N cmbx12XQ cmr12S