Some remarks on S-strongly prime submodules

Let R be a commutative ring with non-zero identity, \(S\subseteq R\) $S\subseteq R$ be a multiplicatively closed subset of R and M be an R-module. A submodule N of M with \((N:_{R}M)\cap S=\emptyset\) $(N{:}_{R}M)\cap S=\mathrm{\varnothing }$ is said to be S-strongly prime, if there exists an \(s\in S\) $s\in S$ such that whenever \(((N+Rx):_{R}M)y\subseteq N\) $((N+Rx){:}_{R}M)y\subseteq N$ , then \(sx\in N\) $sx\in N$ or \(sy\in N\) $sy\in N$ for all \(x,y\in M\) $x,y\in M$ . The aim of this paper is to introduce and investigate some properties of the notion of S-strongly prime submodules, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, Cartesian product. Finally, we state two kind of submodules of the amalgamation module along an ideal and investigate conditions under which they are S-strongly prime.