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Journal ArticleDOI

The Zero-Divisor Graph of a Commutative Ring☆

15 Jul 1999-Journal of Algebra (Academic Press)-Vol. 217, Iss: 2, pp 434-447
TL;DR: For each commutative ring R we associate a simple graph Γ(R) as discussed by the authors, and we investigate the interplay between the ring-theoretic properties of R and the graph-theory properties of Γ (R).
About: This article is published in Journal of Algebra.The article was published on 1999-07-15 and is currently open access. It has received 1087 citations till now. The article focuses on the topics: Zero divisor & Commutative ring.
Citations
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Journal ArticleDOI
TL;DR: In this paper, it was shown that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H, then | G | = | H |, then H is nilpotent.

304 citations

Journal ArticleDOI
TL;DR: In this article, the authors introduced and investigated the total graph of R, denoted by T ( Γ ( R ) ), which is the (undirected) graph with all elements of R as vertices.

290 citations


Cites background from "The Zero-Divisor Graph of a Commuta..."

  • ...In [3], Anderson and Livingston introduced the zero-divisor graph of R, denoted by Γ (R), as the (undirected) graph with vertices Z(R)∗ = Z(R)\{0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, the vertices x and y are adjacent if and only if xy = 0....

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Journal ArticleDOI
TL;DR: In this paper, an undirected graph Γ(S) is associated to each commutative multiplicative semigroup S with 0, where vertices of the graph are labeled by the nonzero zero-divisors of S, and two vertices x,y are connected by an edge in case xy = 0 in S.
Abstract: An undirected graph Γ(S) is associated to each commutative multiplicative semigroup S with 0. The vertices of the graph are labeled by the nonzero zero-divisors of S , and two vertices x,y are connected by an edge in case xy = 0 in S . The properties and possible structures of the graph Γ (S) are studied.

214 citations

Journal ArticleDOI
TL;DR: In this paper, a natural graph associated to the zero-divisors of a commutative ring is considered and the cycle-structure of this graph is classified and some group-theoretic properties of the group of graph-automorphisms are established.
Abstract: There is a natural graph associated to the zero-divisors of a commutative ring In this article we essentially classify the cycle-structure of this graph and establish some group-theoretic properties of the group of graph-automorphisms We also determine the kernel of the canonical homomorphism from to

195 citations

Journal ArticleDOI
TL;DR: For a commutative ring R with set of zero-divisors Z (R), the zero-Divisor graph of R is Γ( R ) = Z ( R )−{0), with distinct vertices x and y adjacent if and only if xy = 0 as mentioned in this paper.

194 citations


Cites background from "The Zero-Divisor Graph of a Commuta..."

  • ...on studying the interplay between graph-theoretic properties of (R) and ring-theoretic properties of R are from [4]....

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References
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Book
01 Jan 1969

16,023 citations

Book
01 May 1997
TL;DR: Gaph Teory Fourth Edition is standard textbook of modern graph theory which covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each chapter by one or two deeper results.
Abstract: Gaph Teory Fourth Edition Th is standard textbook of modern graph theory, now in its fourth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each fi eld by one or two deeper results, again with proofs given in full detail.

6,255 citations

Book
01 Jan 1969
TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

4,168 citations

Book
01 Jun 1974

999 citations

Journal ArticleDOI
Istvan Beck1
TL;DR: In this article, the authors present the idea of coloring of a commutative ring and show that the existence of an infinite clique implies that the clique R = co implies that there exists an infinitely many cliques.

956 citations


"The Zero-Divisor Graph of a Commuta..." refers background in this paper

  • ...This concept is due to Beck [16], who let all the elements of R be vertices and was mainly interested in colorings (also see [2])....

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  • ...The idea of a zero-divisor graph of a commutative ring was introduced w xby I. Beck in 2 , where he was mainly interested in colorings....

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