CS-modules and annihilator conditions

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.