On $n$-absorbing prime ideals of commutative rings

Abstract

This paper investigates the class of rings in which every n-absorbing ideal is a prime ideal, called n-AB ring, where n is a positive integer. We give a characterization of an n-AB ring. Next, for a ring R, we study the concept of Ω(R) = {ω R (I); I is a proper ideal of R}, where ω R (I) = min{n; I is an n-absorbing ideal of R}. We show that if R is an Artinian ring or a Prüfer domain, then Ω(R) ∩ N does not have any gaps (i.e., whenever n ∈ Ω(R) is a positive integer, then every positive integer below n is also in Ω(R)). Furthermore, we investigate rings which satisfy property (**) (i.e., rings R such that for each proper idealwhere M in R (I) denotes the set of prime ideals of R minimal over I). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern n-absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.