On n-absorbing ideals and (m,n)-closed ideals in trivial ring extensions of commutative rings

Abstract

Let [Formula: see text] be a commutative ring with [Formula: see text]. Recall that a proper ideal [Formula: see text] of [Formula: see text] is called a 2-absorbing ideal of [Formula: see text] if [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. A more general concept than 2-absorbing ideals is the concept of [Formula: see text]-absorbing ideals. Let [Formula: see text] be a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is called an n-absorbing ideal of [Formula: see text] 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]. 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] and [Formula: see text] be integers with [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is called an [Formula: see text]-closed ideal of [Formula: see text] if whenever [Formula: see text] for some [Formula: see text] implies [Formula: see text]. Let [Formula: see text] be a commutative ring with [Formula: see text] and [Formula: see text] be an [Formula: see text]-module. In this paper, we study [Formula: see text]-absorbing ideals and [Formula: see text]-closed ideals in the trivial ring extension of [Formula: see text] by [Formula: see text] (or idealization of [Formula: see text] over [Formula: see text]) that is denoted by [Formula: see text].

On $n$-absorbing prime ideals of commutative rings

“…Notice that Ω(R) ⊆ Ω(R ∝ E) from [1,Theorem 5.11(e)]. On the other hand, 0 ∝ E is not an n-absorbing ideal for every positive integer n from [4,Corollary 3.3]. So, ∞ ∈ Ω(R ∝ E).…”

“…If J = 0 ∝ E, then J is a prime ideal (as R is an integral domain) and so ω S (J) = 1. If N is a proper K-subspace of E, the ideal 0 ∝ N is a 2-absorbing ideal of R ∝ E by [4,Theorem 3.3]. Therefore, n = 2.…”

N Sequence Prime Ideals

“…On the other hand, the ideal (5) of is a weakly prime, weakly 2-absorbing, weakly irreducible, 2absorbing, almost prime, strongly 2-irreducible, and almost prime ideal, but it is not an nsequence prime ideal for each . Definition 3.12 [9]. Let with .…”

(m, n)-Closed Submodules of Modules over Commutative Rings

“…The n-absorbing (n ≥ 1) structures have received a significant amount of attention; see for instance [2,4,5,7,24].…”