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 d...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 →...展开更多
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)
文摘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 →...
