scholarly journals On 1-absorbing primary ideals of commutative rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050111 ◽  
Author(s):  
Ayman Badawi ◽  
Ece Yetkin Celikel
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Primary Ideals ◽  
Nonzero Identity

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]

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

2019 ◽  
Vol 19 (10) ◽  
pp. 2050199
Author(s):  
Mohammed Issoual ◽  
Najib Mahdou ◽  
Moutu Abdou Salam Moutui
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Ring Homomorphism ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula

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.

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

2019 ◽  
Vol 18 (07) ◽  
pp. 1950123 ◽  
Author(s):  
Ayman Badawi ◽  
Mohammed Issoual ◽  
Najib Mahdou
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
General Concept ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ring Extension ◽  
Text Recall ◽  
Ideal Formula

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].

scholarly journals On 1-absorbing δ-primary ideals

2021 ◽  
Vol 29 (3) ◽  
pp. 135-150
Author(s):  
Abdelhaq El Khalfi ◽  
Najib Mahdou ◽  
Ünsal Tekir ◽  
Suat Koç
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Basic Properties ◽  
New Class ◽  
Ring Extensions ◽  
Primary Ideals ◽  
Nonzero Identity

Abstract Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

On 1-absorbing prime ideals of commutative rings

2020 ◽  
pp. 2150175
Author(s):  
A. Yassine ◽  
M. J. Nikmehr ◽  
R. Nikandish
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Valuation Domains ◽  
Prufer Domains ◽  
Prüfer Domains

Let [Formula: see text] be a commutative ring with identity. In this paper, we introduce the concept of [Formula: see text]-absorbing prime ideals which is a generalization of prime ideals. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-absorbing prime if for all nonunit elements [Formula: see text] such that [Formula: see text], then either [Formula: see text] or [Formula: see text]. Some properties of [Formula: see text]-absorbing prime are studied. For instance, it is shown that if [Formula: see text] admits a [Formula: see text]-absorbing prime ideal that is not a prime ideal, then [Formula: see text] is a quasi–local ring. Among other things, it is proved that a proper ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing prime if and only if the inclusion [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text] implies that [Formula: see text] or [Formula: see text]. Also, [Formula: see text]-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

Note on strongly quasi-primary ideals

2021 ◽  
pp. 2250201
Author(s):  
Ibtesam Alshammari ◽  
Rania Kammoun ◽  
Abdellah Mamouni ◽  
Mohammed Tamekkante
Keyword(s):  
Commutative Ring ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Ideal Formula ◽  
Primary Ideals

Let [Formula: see text] be a commutative ring with [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then either [Formula: see text] or [Formula: see text]. In this paper, we characterize Noetherian and reduced rings over which every (respectively, nonzero) proper ideal is strongly quasi-primary. We also characterize ring over which every strongly quasi primary ideal of [Formula: see text] is prime. Many examples are given to illustrate the obtained results.

On n-irreducible ideals of commutative rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050120
Author(s):  
Nabil Zeidi
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Ideal Formula ◽  
Strongly Irreducible

Let [Formula: see text] be a commutative ring with [Formula: see text] and [Formula: see text] a positive integer. The main purpose of this paper is to study the concepts of [Formula: see text]-irreducible and strongly [Formula: see text]-irreducible ideals which are generalizations of irreducible and strongly irreducible ideals, respectively. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-irreducible (respectively, strongly [Formula: see text]-irreducible) if for each ideals [Formula: see text] of [Formula: see text], [Formula: see text] (respectively, [Formula: see text]) implies that there are [Formula: see text] of the [Formula: see text]’s whose intersection is [Formula: see text] (respectively, whose intersection is in [Formula: see text]).

On weakly 2-absorbing δ-primary ideals of commutative rings

2020 ◽  
Vol 27 (4) ◽  
pp. 503-516
Author(s):  
Ayman Badawi ◽  
Brahim Fahid
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Primary Ideal ◽  
New Class ◽  
Primary Ideals

AbstractLet R be a commutative ring with {1\neq 0}. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever {a,b,c\in R} and {0\not=abc\in I}, then {ab\in I} or {ac\in\sqrt{I}} or {bc\in\sqrt{I}}. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let {I(R)} be the set of all ideals of R and let {\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever {L,I,J} are ideals of R with {J\subseteq I}, then {L\subseteq\delta(L)} and {\delta(J)\subseteq\delta(I)}. Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., {I\not=R}) is called a weakly 2-absorbing δ-primary ideal if {0\not=abc\in I} implies {ab\in I} or {ac\in\delta(I)} or {bc\in\delta(I)}. For example, let {\delta:I(R)\rightarrow I(R)} such that {\delta(I)=\sqrt{I}}. Then δ is an expansion function of ideals of R, and hence a proper ideal I of R is a weakly 2-absorbing primary ideal of R if and only if I is a weakly 2-absorbing δ-primary ideal of R. A number of results concerning weakly 2-absorbing δ-primary ideals and examples of weakly 2-absorbing δ-primary ideals are given.

Trivial extensions defined by 2-absorbing-like conditions

2018 ◽  
Vol 17 (11) ◽  
pp. 1850208 ◽  
Author(s):  
Mohammed Issoual ◽  
Najib Mahdou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Ideal Formula ◽  
Ring Extensions

Let [Formula: see text] be a commutative ring with [Formula: see text] The notions of 2-absorbing ideal and 2-absorbing primary ideal are introduced by Ayman Badawi as generalizations of prime ideal and primary ideal, respectively. A proper ideal [Formula: see text] of [Formula: see text] is called a 2-absorbing ideal of [Formula: see text] (respectively, 2-absorbing primary ideal) if whenever [Formula: see text] with [Formula: see text] then [Formula: see text] or [Formula: see text] or [Formula: see text] (respectively, [Formula: see text] or [Formula: see text] or [Formula: see text]). In this paper, we investigate the transfer of 2-absorbing-like properties to trivial ring extensions.

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

scholarly journals On 2-absorbing ideals of commutative rings

2007 ◽  
Vol 75 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Maximal Ideals ◽  
Noetherian Domain ◽  
Valuation Domains

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

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