On weakly S-prime submodules

TL;DR

This paper introduces the concept of weakly S-prime submodules in modules over commutative rings, exploring their properties, characterizations, and behavior under various module operations, especially in multiplication modules.

Contribution

It defines weakly S-prime submodules, provides their properties and characterizations, and studies their behavior under module homomorphisms, localizations, quotients, and other constructions.

Findings

Characterization of weakly S-prime submodules in multiplication modules

Behavior of weakly S-prime submodules under module homomorphisms and localizations

Conditions for submodules to be weakly S-prime in amalgamation modules

Abstract

Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\phi$ to be weakly $S$-prime if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in(N:_{R}M)$ or $sm\in N$. Many properties, examples and characterizations of weakly $S$-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly $S$-prime.