; TeX output 2001.11.20:09277 V6N cmbx12MathematischesInstitutmWS2001/026derUniversitXatM`unchenSet66Prof.Dr.B.Pareigis;KNff cmbx12ProblemffsetforAdvancedffAlgebra=}BXQ cmr12(21)\$Letg cmmi12Rn:=URKF!", cmsy10KFforKa eld.`(a)tPShorwthatP:=fLf(a;0)ja2Kܞgisa nitelygeneratedpro-tPjectivreRJ-moSdule.`(b)tPDecide8iftheRJ-moSdulesPqandQ:=f(0;a)ja2Kܞg8aretPisomorphic?aU(c)tPFindadualbasisforPƹ.ʍB(22)`(a)tPLet1RKbSearingandP2cmmi8R %ebeanRJ-module._ShorwthatPӘistPa nitelygeneratedprojectivremoSduleifandonlyifPgӹisatPdirectsummandoftheRJ-moSduleR2n.`(b)tPLetQPR ExbSea nitelygeneratedprojectivrerightRJ-moSdule.tPShorw:thatPƟ2 K cmsy8 >*=`HomUR#I|(PS:;RJ:)isa nitelygeneratedpro-tPjectivreleftRJ-moSdule.B(23)\$Let R#bSearing.ShorwthatforeachprojectiveRJ-moSduleP\$thereisafreeRJ-moSduleFnsucrhthatPLnFP԰=Fƹ.B(24)\$LetCbSeacategorywith niteproducts.6FVoreacrhobjectA\$inCMshorwthatthereexistsamorphismA :4ANC!AA\$satisfyingp|{Ycmr81A 36=UR1A=p2A.:Shorwthatthisde nesanatural\$transformation.8Whatarethefunctors?=}6Duedate:8TVuesdary,27.11.2001,16:15inLectureHall138*;7  K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8Nff cmbx12N cmbx12XQ cmr12