分次环上的分次w-模

Graded w-modules over Graded Rings

R=■σ∈GRσ是有单位元1的交换的G-分次环(在G不需言明时就称R为分次环),并且引入了分次环上的分次w-模等相关概念.证明了:1)设J是R的有限生成分次理想,则J∈GVgr(R)当且仅当J∈GV(R);2)设M是分次模,σ∈G.若M是分次GV-无挠模(或分次GV-挠模),则M(σ)也是分次GV-无挠模(或分次GV-挠模);3)设M是分次模,且是w-模,N是M的分次子模,则N是分次w-模当且仅当N是w-模.特别地,R中的任何分次w-理想都是w-理想.

In this paper, R=■σ∈G R σ is a commutative G -graded ring with identity 1 . We also call R a graded ring for short. Besides, graded w -modules and other related conceptions over a graded ring R are introduced. It is shown that: 1) let J be a finitely generated graded ideal of R . Then J is a graded GV-ideal if and only if J is a GV-ideal. 2) If M is a graded GV-torsion-free module (respectively, GV-torsion module), then the σ-suspended graded module M(σ) is also a graded GV-torsion-free module (respectively, GV-torsion module). 3) Let M be a graded w- module and N be a graded submodule of M . Then N is a graded w -module if and only if N is a w -module. Especially, a graded w - ideal of R is a w -ideal.