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Abstract
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Let R be a commutative ring and let M be an arbitrary R-module. In this talk, we
will introduce the concept of the strongly irreducible submodules of M and we will prove some
properties of them, whenever M is either an arithmetical or a Noetherian module. In the case
when R is Noetherian and M is nitely generated, several characterizations of strongly irre-
ducible submodules are included. Among other things, it is shown that when N is a submodule
of M such that N :R M is not a prime ideal, then N is strongly irreducible if and only if there
exist submodule L of M and prime ideal p of R such that N is p-primary, N L pM and for
all submodules K of M either K N or Lp Kp. In addition, we show that a submodule N
of M is strongly irreducible if and only if N is primary, Mp is arithmetical and N = (pM)(n) for
some integer n > 1, where p = Rad(N :R M) with p 62 AssRR=AnnR(M) and pM is not subset
of 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 :R M is not prime ideal if and only
if there is a prime ideal p of R with pM is not subset of N and Mp is an arithmetical Rp-module.
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