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.