scholarly journals On Weakly δ-Semiprimary Ideals of Commutative Rings

2018 ◽  
Vol 25 (03) ◽  
pp. 387-398
Author(s):  
Ayman Badawi ◽  
Deniz Sonmez ◽  
Gursel Yesilot
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
New Class

Let R be a commutative ring with 1 ≠ 0. A proper ideal I of R is a semiprimary ideal of R if whenever a, b ϵ R and ab ϵ I, we have [Formula: see text] or [Formula: see text]; and I is a weakly semiprimary ideal of R if whenever a, b ϵ R and 0 ≠ ab ϵ I, we have [Formula: see text] or [Formula: see text]. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let I(R) be the set of all ideals of R and let [Formula: see text] 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. Then a proper ideal I of R is called a δ-semiprimary (weakly δ-semiprimary) ideal of R if ab ϵ I (0 ≠ ab ϵ I) implies a ϵ δ(I) or b ϵ δ(I). A number of results concerning weakly δ-semiprimary ideals and examples of weakly δ-semiprimary ideals are given.

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.

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.

Bounds for the Genus of Generalized Total Graph of a Commutative Ring

2019 ◽  
Vol 26 (03) ◽  
pp. 519-528
Author(s):  
T. Asir ◽  
K. Mano
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Total Graph

Let R be a commutative ring with non-zero identity and I its proper ideal. The total graph of R with respect to I, denoted by T (ΓI (R)), is the undirected graph with all elements of R as vertices, and where distinct vertices x and y are adjacent if and only if [Formula: see text]. In this paper, some bounds for the genus of T(ΓI(R)) are obtained. We improve and generalize some results for the total graphs of commutative rings. In addition, we obtain an isomorphism relation between two ideal based total graphs.

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]

scholarly journals On weakly S-prime ideals of commutative rings

2021 ◽  
Vol 29 (2) ◽  
pp. 173-186
Author(s):  
Fuad Ali Ahmed Almahdi ◽  
El Mehdi Bouba ◽  
Mohammed Tamekkante
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
New Class ◽  
Weakly Prime Ideals

Abstract Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.

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 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 (m,n)-closed ideals of commutative rings

2017 ◽  
Vol 16 (01) ◽  
pp. 1750013 ◽  
Author(s):  
David F. Anderson ◽  
Ayman Badawi
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Closed Ideal ◽  
Proper Ideal ◽  
Text Recall ◽  
Positive Integers

Let [Formula: see text] be a commutative ring with [Formula: see text], and let [Formula: see text] be a proper ideal of [Formula: see text]. Recall that [Formula: see text] is an [Formula: see text]-absorbing ideal if whenever [Formula: see text] for [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text]. We define [Formula: see text] to be a semi-[Formula: see text]-absorbing ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. More generally, for positive integers [Formula: see text] and [Formula: see text], we define [Formula: see text] to be an [Formula: see text]-closed ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. A number of examples and results on [Formula: see text]-closed ideals are discussed in this paper.

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.

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]).

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