Weakly classical prime submodules

TL;DR

This paper introduces the concept of weakly classical prime submodules in modules over commutative rings, generalizing classical prime submodules by relaxing the conditions to include nonzero products.

Contribution

The paper defines weakly classical prime submodules, expanding the theory of prime submodules in module theory with a new, less restrictive class.

Findings

Weakly classical prime submodules generalize classical prime submodules.

Characterizations of weakly classical prime submodules are provided.

Connections to existing prime submodule concepts are explored.

Abstract

In this paper, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m \in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$ or $bm\in N$. We introduce the concept of "weakly classical prime submodules." A proper submodule $N$ of $M$ is a weakly classical prime submodule if whenever $a,b\in R$ and $m\in M$ with $0\neq abm\in N$, then $am\in N$ or $bm\in N$.