On weakly S-primary submodules

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