On classical 1-absorbing prime submodules

TL;DR

This paper introduces and explores the concept of classical 1-absorbing prime submodules in modules over commutative rings, providing properties, examples, and characterizations related to tensor products, amalgamated duplications, and local rings.

Contribution

It defines classical 1-absorbing prime submodules and investigates their properties, examples, and behavior under tensor products, amalgamated duplication, and in local rings.

Findings

Characterization of classical 1-absorbing prime submodules

Examples illustrating the concept

Connections with classical prime and 2-absorbing submodules

Abstract

In this study, we aim to introduce the concept of classical 1-absorbing prime submodules of a nonzero unital module $M$ over a commutative ring $A$ with unity. A proper submodule $P$ of $M$ is said to be a classical 1-absorbing prime submodule, if for each $m\in M$ and nonunits $a,b,c\in A,$ $abcm\in P$ implies that $abm\in P$ or $cm\in P$. We give many examples and properties of classical 1-absorbing prime submodules. Also, we investiage the classical 1-absorbing prime submodules of tensor product $F\otimes M$ of a (faithfully) flat $A$-module $F$ and any $A$-module $M$. Furthermore, we determine classical prime, classical 1-absorbing prime and classical 2-absorbing submodules of amalgamated duplication $M\bowtie I$ of an $A$-module $M$ along an ideal $I$. Also, we characterize local rings $(A,\mathfrak{m})$ with $\mathfrak{m}^{2}=0$ in terms of classical 1-absorbing prime submodules.