$\phi$-classical prime submodules

TL;DR

This paper introduces the concept of $\phi$-classical prime submodules in modules over commutative rings, generalizing classical prime submodules by incorporating a function $\phi$ that modifies the prime condition.

Contribution

It defines $\phi$-classical prime submodules and explores their properties, extending the classical notion of prime submodules with a new $\phi$-based framework.

Findings

Introduces $\phi$-classical prime submodules concept.

Establishes properties and characterizations of these submodules.

Provides conditions under which they generalize classical prime submodules.

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 $bm\in N$. Let $\phi:S(M)\to S(M)\cup{\emptyset}$ be a function where $S(M)$ is the set of all submodules of $M$. We introduce the concept of "$\phi$-classical prime submodules". A proper submodule $N$ of $M$ is a $\phi$-classical prime submodule if whenever $a,b\in R$ and $m\in M$ with $abm\in N\backslash\phi(N)$, then $am\in N$ or $bm\in N$.