交换环上的极大性内射模

Max-injective Modules over Commutative Rings

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.