Weakly J-submodules of Modules over Commutative Rings

TL;DR

This paper introduces and explores the concept of weakly J-submodules in modules over commutative rings, generalizing existing J-submodules and analyzing their properties, especially in multiplication modules.

Contribution

It defines weakly J-submodules, extending the notion of J-submodules, and investigates their properties and characterizations in the context of modules over commutative rings.

Findings

Characterization of weakly J-submodules

Properties of weakly J-submodules in multiplication modules

Generalization of J-submodules

Abstract

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. By $J(R),$ we denote the Jacobson radical of $R$. The purpose of this paper is to introduce the concept of weakly $J$-submodules generalizing $J$-submodules. We call a proper submodule $N$ of $M$ a weakly $J$-submodule if whenever $0\neq rm\in N$ for $r\in R$ and $m\in M$, then $r\in(J(R)M:M)$ or $m\in N.$ Various properties and characterizations of weakly $J$-submodules are investigated especially in the case of multiplication modules.