摘要
Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_R^1(R^((α))/R^((β))A, M) = 0 if and only if M_α/r_(M_α)(R^((β))A) ≈ Hom_R(R^((β))A,M) ifand only if r_(M_β)l_(R^((β)))(A) = AM_α. Thus, the notion of (m,n)-injectivity was extended.Moreover, ( α, β) -flatness was characterized via annihilators of matrices, factorizations ofhomomorphisms as well as homological groups so that (m, n)-flat modules, f-projective modules andn-projective modules were consolidated under the notion of (α, β)-flat modules. Furthermore, acharacterization of left R-ML modules and some equivalent conditions for R^((β)) to be left R-MLwere presented. Consequently, the notions of coherent rings, (m, n)-coherent rings and π-coherentrings were consolidated under that of (α, β)-coherent rings.
用环R上的矩阵研究了R 模的一些同调性质.对于任给的基数α,β以及β×α行有限矩阵A,证明了Ext1R(R(α) /R(β)A,M)=0当且仅当Mα/rMα(R(β)A) HomR(R(β)A,M)当且仅当rMβlR(β) (A)=AMα,进一步推广了(m,n) 内射性的概念,并从矩阵的零化子,同态的分解和同调群等角度给出(α,β) 平坦性的等价刻画,从而使(m,n) 平坦模,f 投射模和n 投射模统一到(α,β) 平坦模的概念之下.此外还给出了左R ML模的一个刻画和R(β)A是左R ML模的等价条件,从而把凝聚环、(m,n) 凝聚环、π凝聚环等概念统一到(α,β) 凝聚环的概念之下.
基金
TheFoundationofGraduateCreativeProgramofJiangsu(No.xm04 10),theTeachingandResearchAwardProgramforOutstandingYoungTeachersinHigherEducationInstitutionsofMOE,P.R.C.
