交换环上的极大性内射模  

Max-injective Modules over Commutative Rings

作　　者：王芳贵[1] 汪明义[2] 杨立英[3] 

机构地区：[1]四川师范大学数学学院,四川成都610066 [2]宜宾学院数学学院,四川宜宾643100 [3]广西师范学院数学与计算机科学系,广西南宁530001

出　　处：《四川师范大学学报（自然科学版）》2010年第3期277-285,共9页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(10671137);教育部博士点专项科研基金(20060636001)资助项目

摘　　要：设R是交换环,■表示R的极大理想生成的乘法系,M是R-模.若对R的任何极大理想m,有ExtR1(R/m,M)=0,则M称为极大性内射模.若R自身为极大性内射模,则R称自极大性内射环.若对J∈■,x∈M,由Jx=0能推出x=0,则M称为■-无挠模.证明了在Dedekind整环上,M是极大性内射模当且仅当M是内射模.指出若R的极大理想都是有限生成的,则每个■-无挠模存在极大性内射包络.还证明了若R是■-无挠的自极大性内射模,则自反模是极大性内射模,且非极大素理想都是极大性内射模;若还有R的每个极大理想是有限生成的,则自由模与投射模是极大性内射模.最后,证明了在MFG整环上,平坦模是极大性内射模.Let R be a commutative ring and be the multiplicative system generated by maximal ideals of R.An R-module M is called max-injective if Ext1R（R/m,M）=0 for every maximal ideal m of R.A ring R is called self max-injective if R is max-injective.M is called -torsion-free if Jx=0 for J∈ and x∈M implies x=0.It is shown in this paper that if R is a Dedekind domain,then M is max-injective if and only if M is injective and that if every maximal ideal of R is finitely generated,then -torsion-free modules have the max-injective hulls.It is also shown that if R is an -torsion-free self max-injective ring,then reflexive modules and non-maximal prime ideals are max-injective;moreover,if every maximal ideal of R is finitely generated,then projective modules are max-injective.Finally,it is shown that flat modules over MFG domains are max-injective.

关 键 词：■-无挠模 极大性内射模 自极大性内射环 

分 类 号：O154[理学—数学]