2-irreducible and strongly 2-irreducible ideals of commutative rings

TL;DR

This paper introduces and studies the properties of 2-irreducible and strongly 2-irreducible ideals in commutative rings, generalizing existing notions of irreducibility to more complex ideal intersections.

Contribution

It defines the concepts of 2-irreducible and strongly 2-irreducible ideals, extending the theory of irreducible ideals in commutative algebra.

Findings

Characterization of 2-irreducible ideals

Characterization of strongly 2-irreducible ideals

Relations between these new ideals and existing classes

Abstract

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.