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.

Noetherian Rings in Which Every Ideal is a Product of Primary Ideals

1980 ◽  
Vol 23 (4) ◽  
pp. 457-459 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Number Theory ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Primary Ideals

The classical rings of number theory, Dedekind domains, are characterized by the property that every ideal is a product of prime ideals. More generally, a commutative ring R with identity has the property that every ideal is a product of prime ideals if and only if R is a finite direct sum of Dedekind domains and special principal ideal rings. These rings, called general Z.P.I. rings, are also characterized by the property that every (prime) ideal is finitely generated and locally principal.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

Graded 1-absorbing prime ideals of graded commutative rings

2021 ◽  
Author(s):  
Mohammed Issoual
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Graded Ring

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

scholarly journals On Graded 2-Prime Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 493
Author(s):  
Malik Bataineh ◽  
Rashid Abu-Dawwas
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Principal Ideal Domain ◽  
Graded Rings ◽  
Graded Ring ◽  
Ideal Domain ◽  
Primary Ideals ◽  
Weakly Prime Ideals

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ideals and graded weakly prime ideals are also different. Several properties of graded 2-prime ideals are investigated. We study graded rings in which every graded 2-prime ideal is graded prime, we call such a graded ring a graded 2-P-ring. Moreover, we introduce the concept of graded semi-primary ideals, and show that graded 2-prime ideals and graded semi-primary ideals are different concepts. In fact, we show that graded semi-primary, graded 2-prime and graded primary ideals are equivalent over Z-graded principal ideal domain.

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 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-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].

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.

Annihilating-ideal graphs of commutative rings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050121
Author(s):  
M. Aijaz ◽  
S. Pirzada
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Metric Dimension ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with unity [Formula: see text]. The annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is defined to be the graph with vertex set [Formula: see text] — the set of non-zero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] adjacent if and only if [Formula: see text]. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of [Formula: see text] in terms of their degrees as vertices of [Formula: see text]. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

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