S-prime and S-weakly prime submodules

Emel Aslankarayigit Ugurlu

paper

S-prime and S-weakly prime submodules

arXiv:2005.08733

Abstract

In this study, all rings are commutative with non-zero identity and all modules are considered to be unital. Let $M$ be a left $R$ -module. A proper submodule $N$ of $M$ is called an $S$ - $w e ak l y$ $p r im e$ submodule if $0_{M} \neq = f (m) \in N$ implies that either $m \in N$ or $f (M) \subseteq N,$ where $f \in S = E n d (M)$ and $m \in M$ . Some results concerning $S$ -prime and $S$ -weakly prime submodules are obtained. Then we study $S$ -prime and $S$ -weakly prime submodules of multiplication modules. Also for $R$ -modules $M_{1}$ and $M_{2},$ we examine $S$ -prime and $S$ -weakly prime submodules of $M = M_{1} \times M_{2},$ where $S = S_{1} \times S_{2},$ $S_{1} = E n d (M_{1})$ and $S_{2} = E n d (M_{2})$ .