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

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) is an element of R and a(1)...a(m) is an element of I, then a(1)...a(n) is an element of I or a(n+1)...am is an element of 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) ? a(m) is an element of I for some elements a(1), ..., a(m) is an element of R, then there are n of the ai'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) ? a(m) is an element of I for some nonunit elements a(1), ..., a(m) is an element of R , then there are n of the ai'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.