# Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

```@article{Anderson2003ZerodivisorGV,
title={Zero-divisor graphs, von Neumann regular rings, and Boolean algebras},
author={David F. Anderson and Ronnie Levy and Jay Shapiro},
journal={Journal of Pure and Applied Algebra},
year={2003},
volume={180},
pages={221-241}
}```
• Published 15 May 2003
• Mathematics
• Journal of Pure and Applied Algebra

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• 2017
. Let R be a commutative ring with zero-divisors Z ( R ). The extended zero-divisor graph of R , denoted by Γ( R ), is the (simple) graph with vertices Z ( R ) ∗ = Z ( R ) \{ 0 } , the set of nonzero

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• Mathematics
• 2006
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• Mathematics
• 2014
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## References

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• Mathematics
• 2002
Let R be a commutative ring, and let ZðRÞ denote its set of zerodivisors. We associate a simple graph GðRÞ to R with vertices ZðRÞ 1⁄4 ZðRÞ f0g, the set of nonzero zero-divisors of R. Two distinct

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• Mathematics
• 1993
A commutative ring R can be considered as a simple graph whose vertices are the elements of R and two different elements x and y of R are adjacent if and only if xy = 0. Beck conjectured that χ(R) =

### Structure in the Zero-Divisor Graph of a Non-Commutative Ring

In a manner analogous to the commutative case, the zero-divisor graph of a noncommutative ring R can be defined as the directed graph G. It has been shown that G is not a tournament if R is a finite

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• Mathematics
• 2002
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

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• Mathematics
• 1999
For each commutative ring R we associate a (simple) graph Γ(R). We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ(R).

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This book is an introduction to the theory of associative rings and their modules, designed primarily for graduate students. The standard topics on the structure of rings are covered, with a

### CYCLES AND SYMMETRIES OF ZERO-DIVISORS

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