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 ∈ R and a 1 . . . a m ∈ I , then a 1 . . . a n ∈ I or a n + 1 . . . a m ∈ 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 ∈ I for some elements a 1 , . . . , a m ∈ 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 ⋯ a m ∈ I for some nonunit elements a 1 , . . . , a m ∈ 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.