自内射环族的矩阵环  

Matrix Rings over a Family of Self-injective Rings

作　　者：张圣贵[1] 

机构地区：[1]福建师范大学数学系,福建福州350007

出　　处：《福建师范大学学报（自然科学版）》2000年第3期1-6,共6页Journal of Fujian Normal University：Natural Science Edition

基　　金：福建省自然科学基金资助项目!( A970 1 4 )

摘　　要：主要证明以下结论 :( 1 )若 {αβα|α,β∈ B}是右忠实的双模同态族 ,则 R是局部左自内射环当且仅当 {Rα}α∈ A是左自内射环族且对任意α,β∈ A,μαβ∶ Rαβ→ Hom Rβ( Rβα,Rβ)是同构当且仅当对 A的任意非空有限子集 B,作为左 e BRe B-模 ,有 e BRe B e BE( R) e B;( 2 )若 {αβα|α,β∈ A}是右忠实的双模同态族 ,{ββγ|β,γ∈A}是左忠实的双模同态族 ,则 R是局部左 PF环当且仅当每一个 Rα都是左 PF环 ,α∈ A;( 3 )若 {αβα,αββ|α,β∈ A}是双模同构族 ,则 ( i) R是局部左 Artin环 (局部左 Noether环 )当且仅当 Rα是左 Artin环 (左 Noether环 ) ,α∈ A;( ii) R是局部 QF环当且仅当 {Rα}α∈ A是 QF环族且对任意α,β∈ A,μαβ∶ Rαβ→ Hom Rβ( Rβα,Rβ)Proves the following results:(1) If ｛ αβα ｜α,β∈B｝ is a family of right faithful bimodule homomorphisms, then R is a local left self injective ring if and only if ｛R α｝ α∈A  is a family of left self injective rings and for every α,β∈A,μ αβ ∶R αβ → Hom R β (R βα ,R β) is an isomorphism if and only if for every nonemty subset B of A ,  e BRe Be B(R)e B as a left e BRe B module .(2) If ｛ αβα ｜α,β∈A｝ is a family of right faithful bimodule homomorphisms,｛ ββγ ｜β,γ∈A｝ is a family of left faithful bimodule homomorphisms, then R is a local left PF ring if and only if R α is a left PF ring for every α∈A .(3)If ｛ αβα , αββ ｜α,β∈A｝ is a family of bimodule isomomorphisms, then(i) R is a local left Artinian(Noetherian) ring if and only if R α is a left Artinian(Noetherian) ring for every α∈A.(ii) R is a local QF ring if and only if ｛R α｝ α∈A  is a family of QF rings and for all α,β∈A,μ αβ ∶R αβ → Hom R β (R βα ,R β) is an isomorphism. [WT5HZ]

关 键 词：局部左PF环 局部QF环 自内射环族 矩阵环 

分 类 号：O153.3[理学—数学]