S-内射模及S-内射包络

设R是环.设S是一个左R-模簇,E是左R-模.若对任何N∈S,有Ext_R~1(N,E)=0,则E称为S-内射模.本文证明了若S是Baer模簇,则关于S-内射模的Baer准则成立;若S是完备模簇,则每个模有S-内射包络;若对任何单模N,Ext_R~1(N,E)=0,则E称为极大性内射模;若R是交换环,且对任何挠模N,Ext_R~1(N,E)=0,则E称为正则性内射模.作为应用,证明了每个模有极大性内射包络.也证明了交换环R是SM环当且仅当T/R的正则性内射包e(T/R)是∑-正则性内射模,其中T=T(R)表示R的完全分式环,当且仅当每一GV-无挠的正则性内射模是∑-正则性内射模.

S-Injective Modules and S-Injective Envelopes

Let R be a ring and let S be a family of left modules.A left module E is called S-injective if Ext_R~1(N,E) = 0 for any N∈S.In this paper it is shown that if S is a Baer family of modules,then the Baer Criterion for S-injective modules holds and that if S is a complete family of modules,then every module has an S-injective envelope.A module E is called max-injective if Ext_R~1(N,E) = 0 for any simple module N.A module E over a commutative ring R is called reg-injective if Ext_R~1(N,E) = 0 for any torsion module N.It is shown that every module has a max-injective envelope and that a commutative ring R is an SM ring if and and if the reg-injective envelope e(T/R) of T/R isΣ-reg-injective,where T = T(R) is the total quotient ring of R;if and only if every GV-torsion-free reg-injective module isΣ-reg-injective.