On weakly 2-absorbing ideals of commutative rings

Abstract

Let R be a commutative ring with identity 1 not equal to 0. 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 abc is not equal to 0 for some a, b, c in R and abc is in I, then ab is in I or ac is in I or bc is in I. For example, every proper ideal of a quasi-local ring (R,M) with M3 equals {0} is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal I of R with I3 not equal to {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 M3 equals {0} or R is ring-isomorphic to (R1 × F) where R1 is a quasi-local ring with maximal ideal M such that M2 equals {0} and F is a field or R is ring-isomorphic to (F1 × F2 × F3) for some fields F1, F2, F3.