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 $a_{1}cdots a_{n}min N$ for $a_{1},ldots,a_{n}in R$ and $min M$, then either $a_{1}cdots a_{n}in (N :_R M)$ or there are $n-1$ of $a_{i}$\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

ON WEAKLY 2-ABSORBING IDEALS OF COMMUTATIVE RINGS

Abstract. Let R be a commutative ring with identity 1 ̸ = 0. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R isweakly prime if a, b ∈ R with 0 ̸ = ab ∈ I, then either a ∈ I or b ∈ I. Also a proper ideal I of R is said to be 2-absorbing if whenever a, b, c ∈ R and abc ∈ I, then either ab ∈ I or ac ∈ I or bc ∈ I. In this paper, we introduce the concept of a weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing ideal of R if whenever a, b, c ∈ R and 0 ̸ = abc ∈ I, then either ab ∈ I or ac ∈ I or bc ∈ I. For example, every proper ideal of a quasi-local ring (R, M) with M 3 = {0} is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal I of R with I3 ̸ = 0 is a 2-absorbing ideal of R. We show that every proper ideal of a commutative ring R is a weakly 2-absorbing ideal if and only if either R is a quasi-local ring with maximal ideal M such that M 3 = {0} or R is ringisomorphic to R1 × F where R1 is a quasi-local ring with maximal ideal M such that M 2 = {0} and F is a field or R is ring-isomorphic to F1 × F2 × F3 for some fields F1, F2, F3. 1