# Beck′s Coloring of a Commutative Ring

TL;DR: In this paper, it was shown that if A is a regular Noetherian ring with maximal ideals N 1,..., Ns, such that each A/Ni is finite, then for R = A/Nn11 ··· Nnss, χ(R) = cl(R).

About: This article is published in Journal of Algebra.The article was published on 1993-08-15 and is currently open access. It has received 331 citations till now. The article focuses on the topics: Commutative ring & Noetherian ring.

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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).

1,087 citations

### Cites background from "Beck′s Coloring of a Commutative Ri..."

...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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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

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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

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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 "Beck′s Coloring of a Commutative Ri..."

...The zero-divisor graph of a commutative ring has also been studied in [5,3,9,18,19], and the zero-divisor graph concept has recently been extended to noncommutative rings in [22] and commutative semigroups in [8]....

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TL;DR: In this paper, it was shown that if R is a local commutative ring with at least 33 elements, and Γ ( R )≠∅, then Γ( R ) is not planar.

184 citations

### Cites background from "Beck′s Coloring of a Commutative Ri..."

...In addition, we prove that if R is a ring such that the zero divisor graphΓ (R) is completer-partite, then we have: (1) If R is Artinian, thenR is finite; (2) If R is Noetherian, thenR is a subring of a ringF × S, whereF is a field andS is a finite ring....

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