On Almost ‎‎S‎‎-prime submodules

‎‎Let R be a commutative ring with non-zero identity, S\subseteq R be a multiplicatively closed subset of R and let M be an R-module.  A submodule N of M with (N:_{R}M)\cap S=\emptyset is said to be almost S-prime, if there exists an s\in S such that whenever rm\in N-(N:_{R}M)N, then sm\in N or sr\in (N:_{R}M) for each r\in R, m\in M. The aim of this paper is to introduce and investigate some properties of the notion of almost S-prime submodules, especially in multiplication modules. Moreover, we investigate the behaviour of this structure under module homomorphisms, localizations, quotient modules, Cartesian product. Finally, we state two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are almost S-prime.

Keywords:

S-prime submodule‎ , ‎almost S-prime submodule‎ , ‎multiplication module