1-absorbing primary submodules

TL;DR

This paper introduces the concept of 1-absorbing primary submodules in modules over commutative rings, extending the ideal-based notion and exploring their properties and related theorems.

Contribution

It extends the concept of 1-absorbing primary ideals to submodules, providing new characterizations and an avoidance theorem for these submodules.

Findings

Characterization of 1-absorbing primary submodules

Properties and structure theorems for these submodules

Proof of a 1-absorbing primary avoidance theorem

Abstract

Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule $N$ of $M$ is said to be a 1-absorbing primary submodule if whenever non-unit elements $a,b\in R$ and $m\in M$ with $abm\in N$, then either $ab\in(N:_{R}M)$ or $m\in M-rad(N).$ Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.