关于相对平坦性与相对凝聚性的注记

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1、http://www.paper.edu.cn1ANoteonRelativeFlatnessandCoherenceXiaoxiangZhangJianlongChenDepartmentofMathematicsSoutheastUniversity，Nanjing210096E-mail:z990303@seu.edu.cnAbstractLetRbearingandMafixedrightR-module.AnewcharacterizationofM-flatnessisgivenby

2、certainlinearequations.ForaleftR-moduleFsuchthatthecanonicalmapM⊗RF→HomR(M*,F)isinjective,whereM*=HomR(M,R),theM-flatnessofFischaracterizedviacertainmatrixsubgroups.AnexampleisgiventoshowthatRneednotbeM-coherentevenifeveryleftR-moduleisM-flat.Moreove

3、r,somepropertiesofM-coherentringsarediscussed.Keywords:M-flatmodule,matrixsubgroup，M-coherentring1Introductions[2]Recently,DaunsintroducedthenotionofcoherenceofaringRrelativetoarightR-moduleM.LetRbearing,MafixedrightR-moduleandσ[M]thefullsubcategoryo

4、fthecategoryofrightR-modulessubgeneratedbyM(see[6,p.118]).Recallfrom[2]thataleftR-moduleFisσ[M]-flatifforanyexactsequence0→X→Yinσ[M],thesequence0→X⊗RF→Y⊗RFisexact.Following[6]Wisbauer,FiscalledM-flatifthesequence0→K⊗RF→M⊗RFisexactforeverysubmodule0≤M

5、.[2,Proposition1.6]DaunsprovedthatRFisM-flatifandonlyifitisσ[M]-flat.[2]FollowingDauns,arightR-moduleNisM-coherentifforany0≤A

6、isM-coherent.ThepurposeofthisnoteistoinvestigateM-flatmodulesandM-coherentringsfromsomenewaspects.ThroughoutRisanassociativeringwithidentityandallmodulesareunitary.Forapositivenintegern,R(resp.Rn)denotesthedirectsumofncopiesofRR(resp.RR)whoseelements

7、arewrittenasn“row(resp.column)vectors”.Similarly,MstandsforthedirectsumofncopiesofMR.Foreachm=n(m1,m2,…,mn)∈M,therightannihilatorofminRnissymbolizedbyrR(m),thatis,nnr(m)TTRn={(r1,r2,…,rn)∈Rn

8、(m1,m2,…,mn)(r1,r2,…,rn)=∑miri=0}.i=1⊥Notethatr(m)isnothing

9、morethanmin[2]forallm∈M.GivenanR-moduleP,thedualRn++moduleandthecharactermodulearedenotedbyP*andPrespectively,i.e.,P*=HomR(P,R)andP=HomZ(P,Q/Z),whereQ(resp.Z)istheringofrationalnumbers(resp.integers).1ThisworkissupportedinpartbytheNSFCofChina(No.1057