On S -submodules of modules over commutative rings

Let R be a commutative ring with identity, S be a multiplicatively closed subset of R and M be an R-module. In this paper, we introduce and study the notion of S-submodules as a generalization of S-ideals introduced by Khashan and Hussein 14. We define a proper submodule N of M to be an S-submodule if whenever sm ∈ N for some s ∈ S and m ∈ M, we have m ∈ N. Several properties, characterizations and examples of S-submodules are given. In particular, we characterizes modules whose only S-submodule is the zero submodule. Moreover, we prove that submodules that admit primary decomposition correspond to S-submodules for which S is the complement of a finite union of prime ideals. Finally, we study the transfer of the S-submodule property to some constructions of modules such localization, direct product and amalgamation of modules along an ideal.