CS‐modules and annihilator conditions

Mahmoud A Kamal,Amany M Menshawy

doi:10.1155/s0161171203206219

Abstract

We study S − R‐bimodules SMR with the annihilator condition S = lS(A) + lS(B) for any closed submodule A, and a complement B of A, in MR. Such annihilator condition has a direct connection with the CS‐condition for MR. We make use of this to give a new characterization of CS‐modules. Bimodules SMR for which rMlS(A) = A (for every closed submodule A of MR) are also dealt with. Such modules are called W∗‐modules. We give the extra added annihilator conditions to W∗‐modules to be equivalent to the continuous (quasicontinuous) modules.

Highlights

Let R and S be rings and let S MR be a bimodule
The module M is continuous if it is a CS-module and satisfies condition (C2): if A B ≤ M with A ≤⊕ M, B ≤⊕ M
A generalization of condition (C2) is (GC2): if A is a submodule of M with A M, A ≤⊕ M

Summary

Introduction

Let R and S be rings and let S MR be a bimodule. For any X ≤ M and T ≤ S, write lS (X) = {s ∈ S : sX = 0} and rM (T ) = {m ∈ M : T m = 0}. The following are equivalent: (1) for every closed submodule A of MR, there exists a complement B of A in MR such that S = lS (A) + lS (B); (2) MR is CS and every idempotent of End(MR) is central; (3) MR is CS and every closed submodule of MR is fully invariant.

Results

Conclusion