S-prime and S-weakly prime submodules

TL;DR

This paper introduces and studies the concepts of S-weakly prime and S-prime submodules in modules over commutative rings, exploring their properties, especially in multiplication modules and product modules, expanding the understanding of submodule structures.

Contribution

It defines S-weakly prime and S-prime submodules, investigates their properties, and analyzes their behavior in multiplication and product modules, providing new insights into submodule theory.

Findings

Characterization of S-weakly prime submodules

Results on S-prime submodules in multiplication modules

Analysis of S-prime and S-weakly prime submodules in product modules

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 $S$-$weakly$ $prime$ submodule if $0_{M}\neq f(m)\in N$ implies that either $m\in N$ or $f(M)\subseteq N,$ where $f\in S=End(M)$ 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 $S=S_{1}\times S_{2},$ $S_{1}=End(M_{1})$ and $S_{2}=End(M_{2})$.