On S-strong Mori modules

Let A be an integral domain, S subset of A be a multiplicative set and M a w-module as an A-module. In this paper we investigate S-SM-modules. We give an S-version of the result of Wang and McCasland (Commun Algebra 25:1285-1306, 1997) in the case where S is countable. We prove that M is an S-SM-module if and only if every increasing sequence of w-submodules of M is S-stationary if and only if every increasing sequence of S-w-finite w-submodules of M is S-stationary if and only if every increasing sequence of w-finite type submodules of M is S-stationary.