On weakly S-primary submodules

Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :R M) ? S = ? to be weakly S-primary if there exists s ? S such that whenever a ? R and m ? M with 0 , am ? N, then either sa ??(N :R M) or sm ? N. We present various properties and characterizations of this concept (especially in faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations is investigated. Finally, we determine some conditions under which two kinds of submodules of the amalgamation module along an ideal are weakly S-primary.