Some characterizations of strongly irreducible submodules in arithmetical and Noetherian modules

TL;DR

This paper investigates properties and characterizations of strongly irreducible submodules in arithmetical and Noetherian modules over commutative rings, establishing relationships with other submodule types and providing new criteria.

Contribution

It introduces new characterizations of strongly irreducible submodules, especially in Noetherian modules, and explores their relationships with prime, irreducible, and primal submodules.

Findings

Characterizations of strongly irreducible submodules in Noetherian modules.

Conditions under which submodules are strongly irreducible based on primarity and arithmetical properties.

Existence criteria for strongly irreducible submodules in torsion-free modules over integral domains.

Abstract

The purpose of the present paper is to prove some properties of the strongly irreducible submodules in the arithmetical and Noetherian modules over a commutative ring. The relationship among the families of strongly irreducible submodules, irreducible submodules, prime submodules and primal submodules is proved. Also, several new characterizations of the arithmetical modules are given. In the case when $R$ is Noetherian and $M$ is finitely generated, several characterizations of strongly irreducible submodules are included. Among other things, it is shown that when $N$ is a submodule of $M$ such that $N:_RM$ is not a prime ideal, then $N$ is strongly irreducible if and only if there exist submodule $L$ of $M$ and prime ideal $\frak p$ of $R$ such that $N$ is $\frak p$-primary, $N\subsetneqq L\subseteq \frak pM$ and for all submodules $K$ of $M$ either $K\subseteq N$ or $L_{\frak p}\subseteq K_{\frak p}$. In addition, we show that a submodule $N$ of $M$ is strongly irreducible if and only if $N$ is primary, $M_{\frak p}$ is arithmetical and $N=(\frak pM)^{(n)}$ for some integer $n>1$, where $\frak p=\Rad(N:_RM)$ with $\frak p\not\in \Ass_RR/\Ann_R(M)$ and $\frak pM\nsubseteq N$. As a consequence we deduce that if $R$ is integral domain and $M$ is torsion-free, then there exists a strongly irreducible submodule $N$ of $M$ such that $N:_RM$ is not prime ideal if and only if there is a prime ideal $\frak p$ of $R$ with $\frak pM\nsubseteq N$ and $M_{\frak p}$ is an arithmetical $R_{\frak p}$-module.