  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.
