On n-absorbing and strongly n-absorbing ideals of amalgamation

Abstract

Let [Formula: see text] be a commutative ring with [Formula: see text]. Let [Formula: see text] be a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is called an n-absorbing ideal (respectively, a strongly n-absorbing ideal) of [Formula: see text] as in [D. F. Anderson and A. Badawi, On [Formula: see text]-absorbing ideals of commutative rings, Comm. Algebra 39 (2011) 1646–1672] if [Formula: see text] and [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text] (respectively, if whenever [Formula: see text] for ideals [Formula: see text] of [Formula: see text], then the product of some [Formula: see text] of the [Formula: see text]s is contained in [Formula: see text]). The concept of [Formula: see text]-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of [Formula: see text] is a 1-absorbing ideal of [Formula: see text]). Let [Formula: see text] be a ring homomorphism and let [Formula: see text] be an ideal of [Formula: see text] This paper investigates the [Formula: see text]-absorbing and strongly [Formula: see text]-absorbing ideals in the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect [Formula: see text] denoted by [Formula: see text] The obtained results generate new original classes of [Formula: see text]-absorbing and strongly [Formula: see text]-absorbing ideals.