We study S − R -bimodules S M R with the annihilator condition S = l S ( A ) + l S ( B ) for any closed submodule A , and a complement B of A , in M R . Such annihilator condition has a direct connection with the CS-condition for M R . We make use of this to give a new characterization of CS-modules. Bimodules S M R for which r M l S ( A ) = A (for every closed submodule A of M R ) 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.