PS and CESS Property of Formal Triangular Matrix Rings  

形式三角矩阵环的PS性质和CESS性质(英文)

作　　者：张文汇 刘仲奎 

机构地区：[1]Department of Mathematics, Northwest Normal University

出　　处：《Journal of Mathematical Research and Exposition》2008年第4期981-986,共6页数学研究与评论（英文版）

基　　金：the National Natural Science Foundation of China (No.10171082);TRAPOYT (No.200280);Yong Teachers Research Foundation of NWNU (No.NWNU-QN-07-36)

摘　　要：Let R be a ring. Recall that a right R-module M (RR, resp.) is said to be a PS-module (PS-ring, resp.) if it has projective socle. M is called a CESS-module if every complement summand in M with essential socle is a direct summand of M. We show that the formal triangular matrix ring T = A 0M B is a PS-ring if and only if A is a PS-ring, MA and lB(M) = {b ∈ B | bm = 0,m ∈ M} are PS-modules and Soc(lB(M)) M = 0. Using the alternative of right T-module as triple (X,Y )f with X ∈ Mod-A, Y ∈ Mod-B and f : YM →...Let R be a ring. Recall that a right R-module M (RR, resp.) is said to be a PS-module (PS-ring, resp.) if it has projective socle. M is called a CESS-module if every complement summand in M with essential socle is a direct summand of M. We show that the formal triangular matrix ring T = A 0M B is a PS-ring if and only if A is a PS-ring, MA and lB(M) = {b ∈ B | bm = 0,m∈ M} are PS-modules and Soc(lB(M)) M = 0. Using the alternative of right T-module as triple (X,Y )f with X ∈ Mod-A, Y ∈ Mod-B and f : YM→ X in Mod-A, we show that if TT is a CESS-module, then AA and MA are CESS-modules.

关 键 词：formal triangular matrix ring PS-ring CESS-module. 

分 类 号：O153.3[理学—数学]