15 research outputs found
2-irreducible and strongly 2-irreducible ideals of commutative rings
An ideal I of a commutative ring R is said to be 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 strongly irreducible if for each ideals J, K of R, J\cap
K\subseteq I implies that J\subset I or K\subset 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 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 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.Comment: 15 page
On n-absorbing submodules
All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate n-absorbing submodules. For this reason we introduce the concept of n-absorbing submodules generalizing n-absorbing ideals of rings. Let M be an R-module. A proper submodule N of M is called an n-absorbing submodule if whenever for and , then either or there are of \u27s whose product with m is in N. We study the basic properties of n-absorbing submodules and then we study n-absorbing submodules of some classes of modules (e.g. Dedekind modules, Prüfer modules, etc.) over commutative rings