On 1-absorbing primary ideals of commutative rings

Abstract

<jats:p> Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text] </jats:p>

Note on weakly 1-absorbing primary ideals

“…Suppose that R is a domain. Let 0 P be a non-maximal prime ideal of R. Then, PM is a 1-absorbing primary ideal of R and √ PM = P (by [4,Theorem 7]). Then PM is a weakly primary ideal of R. Let 0 x ∈ P and y ∈ M \ P. We have 0 xy ∈ PM and y P = √ PM.…”

“…A proper ideal I of R is called a weakly primary ideal of R if whenever a, b ∈ R and 0 ab ∈ I, then a ∈ I or b ∈ √ I. Recent generalizations of primary ideals and weakly primary ideals are, respectively, the notions of 1-absorbing primary ideals and weakly 1-absorbing primary ideals introduced by Badawi and Yetkin in [4,6]. A proper ideal I of R is called 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √ I.…”

On 1-absorbing δ-primary ideals

“…[2,Theorem 2.4]. Recently, Badawi and Yetkin [4] consider a new class of ideals called the class of 1-absorbing primary ideals.…”

1-absorbing primary submodules

“…According to [14], a proper submodule N of M is said to be 2-absorbing primary provided that a, b ∈ R, m ∈ M and abm ∈ N imply either ab ∈ (N : R M ) or am ∈ M -rad(N ) or bm ∈ M -rad(N ). As a recent study, the class of 1-absorbing primary ideals was defined in [7].…”

“…According to [7], a proper ideal I of R is said to be a 1-absorbing primary ideal if whenever non-unit elements a, b, c of R and abc ∈ I, then ab ∈ I or c ∈ √ I. Our aim is to extend the notion of 1-absorbing primary ideals to 1-absorbing primary submodules.…”