MMN-1490
2-irreducible and strongly 2-irreducible ideals of commutative rings
Abstract
An ideal $I$ of a commutative ring $R$ is said to be {\it irreducible} if it cannot be
written as the intersection of two larger ideals. A proper ideal $I$ of a ring
$R$ is said to be {\it strongly irreducible} if for each ideals $J,~K$ of $R$,
$J\cap K\subseteq I$ implies that $J\subseteq I$ or
$K\subseteq I$. In this paper, we introduce the
concepts of 2-irreducible and strongly 2-irreducible ideals which are generalizations of
irreducible and strongly irreducible ideals, respectively.
We say that a proper ideal $I$ of a ring $R$ is {\it 2-irreducible} if for each ideals
$J,~K$ and $L$ of $R$, $I=J\cap K\cap L$ implies that either $I=J\cap K$ or
$I=J\cap L$ or $I=K\cap L$. A proper ideal $I$ of a ring $R$ is called {\it strongly 2-irreducible}
if for each ideals $J,~K$ and $L$ of $R$, $J\cap K\cap L\subseteq I$ implies that either $J\cap K\subseteq I$ or
$J\cap L\subseteq I$ or $K\cap L\subseteq I$.
Vol. 17 (2016), No. 1, pp. 441-455
DOI: 10.18514/MMN.2016.1490