Some characterizations of strongly irreducible submodules in arithmetical and Noetherian modules
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.