# On ( m , n )-absorbing ideals of commutative rings

Research paper by BATOOL ZAREI JALAL ABADI, HOSEIN FAZAELI MOGHIMI

Indexed on: 01 Jan '17Published on: 09 Dec '16Published in: Proceedings - Mathematical Sciences

#### Abstract

Abstract Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a 1⋯a m ∈I for a 1,…, a m ∈R∖U(R), then there are n of the a i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m − 1 maximal ideals.AbstractLet R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a 1⋯a m ∈I for a 1,…, a m ∈R∖U(R), then there are n of the a i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m − 1 maximal ideals.RURRmnmnnIRmna1a m mIa1a m mRURna i iImnmnmnnmnm