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

Abstract

Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of 1-absorbing prime ideals. Let m, n be nonzero positive integers such that $m > n$. A proper ideal I of R is said to be an (m, n)-absorbing prime ideal if whenever nonunit elements $a_1,...,a_{m} \in R$ and $a_1...a_{m}\in I$, then $a_1...a_{n} \in I$ or $a_{n+1}...a_m\in I.$ We give some basic properties of this class of ideals and we study (m, n)-absorbing prime ideals of localization of rings, direct product of rings and trivial ring extensions. A proper ideal I of R is called an AB-(m, n)-absorbing ideal of R if whenever $a_1\cdots a_{m} \in I$ for some elements $a_1, ... , a_{m} \in R$, then there are n of the $a_i$’s whose product is in I. A proper ideal I of R is called an (m, n)-absorbing ideal of R if whenever $a_1\cdots a_{m} \in I$ for some nonunit elements $a_1, ... , a_{m} \in R$, then there are n of the $a_i$’s whose product is in I. We study some connections between (m, n)-absorbing prime ideals, (m, n)-absorbing ideals and AB-(m, n)-absorbing ideals of commutative rings.