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
203 Citations
• Mathematics
• 2008
Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y
Let R be a noncommutative ring. The zero-divisor graph of R, denoted by Γ(R), is the (directed) graph with vertices Z(R)∗ = Z(R)− {0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈
• Mathematics
• 2016
In this paper we present a new graph that is closely related to the classical zero-divisor graph. In our case two nonzero distinct zero divisors x and y of a commutative ring R are adjacent whenever
• Mathematics
• 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
• Mathematics
• 2006
ABSTRACT Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ I (R), is the graph whose vertices are the set {x ∈
• Mathematics
• 2014
Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)∖{0} and distinct vertices x and y are adjacent if and only

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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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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) =
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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• 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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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).
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
• Mathematics
• 2002
ABSTRACT Let be a commutative ring. In this paper, we give several divisibility and ring-theoretic conditions for or to be either zero-dimensional or von Neumann regular. We also consider