# Coloring of commutative rings

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.

About: This article is published in Journal of Algebra.The article was published on 1988-07-01 and is currently open access. It has received 956 citations till now. The article focuses on the topics: Principal ideal ring & Von Neumann regular 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 "Coloring of commutative rings"

...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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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

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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 "Coloring of commutative rings"

...This concept is due to Beck [7], who let all the elements of R be vertices and was mainly interested in colorings....

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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 "Coloring of commutative rings"

...The concept of a zero-divisor graph of a commutative ring was introduced by Beck [7]....

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##### References

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01 Jan 1951

TL;DR: In this paper, the Cartesian product of a family of compact sets is shown to be compact, based on TychonofI's theorem, which is a special case of R. RADO's theorem.

Abstract: Our original proof was simplified by SZEKERES. Later, a simple proof, based on TychonofI’s theorem that the Cartesian product of a family of compact sets is compact, was indicated by RABSOX and A. STONE. We suppress these proofs here, since t’heorem 1 can be considered as a special case of a theorem of R. RADO which appeared meanwhile [3], and a topological proof for Rado’s theorem was given by GOTTSCHALK [2].

290 citations