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.