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کاربرد نوع شرط:
- جایگاه : پژوهشی
- مجله: Journal of Algebra and Related Topics
- نوع مقاله: Journal Article
- کلمات کلیدی: diameter,Clique number,girth,Artinian ring,Special principal ideal ring
- چکیده:
- چکیده انگلیسی: The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. Let us denote the collection of all proper ideals of $R$ by $mathbb{I}(R)$ and $mathbb{I}(R)backslash {(0)}$ by $mathbb{I}(R)^{*}$. With $R$, we associate an undirected graph denoted by $g(R)$, whose vertex set is $mathbb{I}(R)^{*}$ and distinct vertices $I_{1}, I_{2}$ are adjacent in $g(R)$ if and only if $I_{1}cap I_{2}neq I_{1}I_{2}$. The aim of this article is to study the interplay between the graph-theoretic properties of $g(R)$ and the ring-theoretic properties of $R$.
- انتشار مقاله: 09-06-1397
- نویسندگان: S. Visweswaran,P. Vadhel
- مشاهده
- جایگاه : پژوهشی
- مجله: Journal of Algebra and Related Topics
- نوع مقاله: Journal Article
- کلمات کلیدی: Clique number,chromatic number,Complete graph,Planar Graph,Comaximal graph of a ring
- چکیده:
- چکیده انگلیسی: The rings considered in this article are commutative with identity. This article is motivated by the work on comaximal graphs of rings. In this article, with any ring $R$, we associate an undirected graph denoted by $G(R)$, whose vertex set is the set of all elements of $R$ and distinct vertices $x,y$ are joined by an edge in $G(R)$ if and only if $Rxcap Ry = Rxy$. In Section 2 of this article, we classify rings $R$ such that $G(R)$ is complete and we also consider the problem of determining rings $R$ such that $chi(G(R)) = omega(G(R))< infty$. In Section 3 of this article, we classify rings $R$ such that $G(R)$ is planar.
- انتشار مقاله: 15-05-1396
- نویسندگان: S. Visweswaran,J. Parejiya
- مشاهده
- جایگاه : پژوهشی
- مجله: Journal of Algebra and Related Topics
- نوع مقاله: Journal Article
- کلمات کلیدی: Maximal non-prime ideal,maximal non-maximal ideal,maximal non-primary ideal,maximal non-irreducible ideal
- چکیده:
- چکیده انگلیسی: The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$, we mean an ideal $I$ of $R$ such that $Ineq R$. We say that a proper ideal $I$ of a ring $R$ is a maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$, $I$ is maximal with respect to the property of being not a prime ideal. The concept of maximal non-maximal ideal and maximal non-primary ideal of a ring can be similarly defined. The aim of this article is to characterize ideals $I$ of a ring $R$ such that $I$ is a maximal non-prime (respectively, a maximal non maximal, a maximal non-primary) ideal of $R$.
- انتشار مقاله: 22-03-1394
- نویسندگان: S. Visweswaran,A. Parmar
- مشاهده
- جایگاه : پژوهشی
- مجله: Algebraic Structures and Their Applications
- نوع مقاله: Journal Article
- کلمات کلیدی: N-prime of $(0)$,B-prime of $(0)$,complement of the annihilating-ideal graph of a commutative ring,vertex cut and cut vertex of a connected graph
- چکیده:
- چکیده انگلیسی: The rings considered in this article are commutative with identity which admit at least two nonzero annihilating ideals. Let $R$ be a ring. Let $mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}$. The annihilating-ideal graph of $R$, denoted by $mathbb{AG}(R)$ is an undirected simple graph whose vertex set is $mathbb{A}(R)^{*}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if $IJ = (0)$. The aim of this article is to classify rings $R$ such that $(mathbb{AG}(R))^{c}$ ( that is, the complement of $mathbb{AG}(R)$) is connected and admits a cut vertex.
- انتشار مقاله: 21-01-1395
- نویسندگان: S. VISWESWARAN,A. PARMAR
- مشاهده