Some Properties of Completely Arithmetical Rings

2016 ◽  
Vol 23 (01) ◽  
pp. 83-88
Author(s):  
Xinmin Lu
Keyword(s):  
Positive Integer ◽  
Proper Ideal ◽  
Strongly Irreducible ◽  
Arithmetical Rings

In this paper, we introduce the concept of completely arithmetical rings and investigate their properties. In particular, we prove that if R is a completely arithmetical ring with J(R)=0, then K0(R) ≅ ℤn for some positive integer n. We also show that such a ring is precisely a ring in which every proper ideal can be written uniquely as a product of finitely many distinct completely strongly irreducible ideals.

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]).

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 STRONGLY IRREDUCIBLE IDEALS

2008 ◽  
Vol 84 (2) ◽  
pp. 145-154 ◽  
Author(s):  
A. AZIZI
Keyword(s):  
Proper Ideal ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Strongly Irreducible

AbstractA proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, $A\cap B \subseteq I$ implies that either $A \subseteq I$ or $B \subseteq I$. In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace topologies.

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.

scholarly journals Studying Semigroups Using the Properties of Their Prime m-Ideals

2020 ◽  
Vol 34 ◽  
pp. 109-125
Author(s):  
M. Munir ◽  
◽  
N. Kausar ◽  
B. Davvaz ◽  
M. Gulistan ◽  
...  
Keyword(s):  
Positive Integer ◽  
Strongly Irreducible

In this article, we present the idea of m-ideals, prime m-ideals and their associated types for a positive integer m in a semigroup. We present different chrarcterizations of semigroups through m-ideals. We demonstrate that the ordinary ideals, and their relevent types differ from the m-ideals and their assocated types by presenting concrete examples on the maximal, irreducible and strongly irreducible m-ideals. We conclude from the study that the introduction of the m-ideal will explore new fields of studies in semigroups and their applications.

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].

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

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.

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