分次投射盖和交换分次完全环  

Graded Projective Covers and Commutative Graded Perfect Rings

作　　者：谢雅静 王芳贵[1] 吴小英[1] XIE Yajing;WANG Fanggui;WU Xiaoying(College of Mathematical Science,Sichuan Normal University,Chengdu 610066,Sichuan)

机构地区：[1]四川师范大学数学科学学院

出　　处：《四川师范大学学报（自然科学版）》2019年第6期729-738,共10页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(11671283)

摘　　要：设G是交换群,R=σ∈G Rσ是交换G-分次环.给出了交换分次半完全环与分次完全环的一些等价刻画.证明:1)分次局部环上任何有限生成分次模有分次投射盖.2)R是分次半完全环当且仅当R是有限个分次局部环的直积.3)R是分次完全环当且仅当R/J g(R)是分次半单环,且每个非零分次模都有极大分次子模;当且仅当每个分次模有关于分次循环子模的降链条件;当且仅当R是分次局部环R i的直积,且每个J g(R i)是T-幂零的.4)若R是强分次环,则R是分次完全环当且仅当R e是完全环.Let G be a commutative group and let R=σ∈G Rσbe a commutative graded ring.In this paper,the equivalent characterizations about graded semiperfect rings and graded perfect rings are given.It is shown that:1)Every finitely generated graded module over a graded local ring has a graded projective cover.2)R is graded semiperfect if and only if R is a direct product of finite graded local rings.3)R is graded perfect if and only if R/J g(R)is graded semisimple and every nonzero graded module has a maximal graded submodule;if and only if every graded module satisfies the descending chain condition on cyclic submodules;if and only if R is a direct product of graded local rings R i,and J g(R i)is a T-nilpotent ideal.4)If R is a strongly graded ring,then R is a graded perfect ring if and only if R e is a perfect ring.

关 键 词：分次多余子模 分次投射盖 分次半完全环 分次完全环 

分 类 号：O153.3[理学—数学] O154[理学—基础数学]