S-内射模及S-内射包络  

S-Injective Modules and S-Injective Envelopes

作　　者：王芳贵[1] 廖家丽[1] 

机构地区：[1]四川师范大学数学与软件科学学院,成都610068

出　　处：《数学学报（中文版）》2011年第2期271-284,共14页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(10671137);教育部博士点专项科研基金(20060636001);四川省重点学科建设基金(SZD0406)资助项目

摘　　要：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.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.

关 键 词：S-内射模 极大性内射模 正则性内射模 

分 类 号：O153.3[理学—数学] O154[理学—基础数学]