On (m,n)-closed ideals of commutative rings

Abstract

Let [Formula: see text] be a commutative ring with [Formula: see text], and let [Formula: see text] be a proper ideal of [Formula: see text]. Recall that [Formula: see text] is an [Formula: see text]-absorbing ideal if whenever [Formula: see text] for [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text]. We define [Formula: see text] to be a semi-[Formula: see text]-absorbing ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. More generally, for positive integers [Formula: see text] and [Formula: see text], we define [Formula: see text] to be an [Formula: see text]-closed ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. A number of examples and results on [Formula: see text]-closed ideals are discussed in this paper.