; TeX output 2001.12.10:17507 V6N cmbx12MathematischesInstitutmWS2001/026derUniversitXatM`unchenSet96Prof.Dr.B.Pareigis;KNff cmbx12ProblemffsetforAdvancedffAlgebra=}BXQ cmr12(33)\$ShorwZthatthereisanaturalisomorphismMap?(g cmmi12AuW!", cmsy10B;Cܞ)PP԰h=\$Mapsu(B;Map+O(A;Cܞ)).DetermineZtheassoSciatedunitandcounit.ʍB(34)\$ShorwthattheunderlyingfunctorU6:URAb0!'oGr59p,thatforgetsthe\$commrutativityaofAbSeliangroups,hasaleftadjoinrtfunctorFc:\$Gro>qX!Ab^.DeterminetheassoSciatedunitandcounit.(Hinrt:\$Considertheconstructionofthecommrutatorfactorgroup.)B(35)\$ShorwhthattheunderlyingfunctorU ¹:Ab/H!/(comMon`from\$theUcategoryofAbSeliangroupstothecategoryofcommrutative\$monoids,/that'forgetstheexistenceofinrverses'ingroups,has\$aleftadjoinrtfunctorF?:.comMon5 -7&!GAbV.EDeterminetheas-\$soSciatedunitandcounit.(Hinrt:mConsidertheconstructionof\$fractionsforrationalnrumbSers.)B(36)\$LetRfɹ:Rɟ!`kS)bSearinghomomorphism. Shorwthatthe\$underlyingfunctorUn:zSF-MoSd$U&!7 lR-_MoSd$*fromthecategory\$ofS׹-moSdulestothecategoryofRJ-modules,[thatforgetstheS׹-\$moSdulestructureonmodulesandreplacesitbrytheRJ-module\$structurercmUR:=fG(rS)m,hasaleftadjoinrtfunctorFc:R-_MoSd\$^ݯ!n3SF-MoSd#andarighrtadjointfunctorG :URR-_MoSd"͐$!4