# 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} }

## 199 Citations

### On the Zero-Divisor Graph of a Ring

- 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…

### Zero-Divisor Graph of Triangular Matrix Rings over Commutative Rings 1

- Mathematics
- 2011

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 ∈…

### On the extended zero divisor graph of commutative rings

- 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…

### EXTENDED ZERO-DIVISOR GRAPHS OF IDEALIZATIONS

- 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…

### Zero-Divisor Graph with Respect to an Ideal

- 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 ∈…

### Some Remarks on the Graph Γ I (R)

- 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…

## References

SHOWING 1-10 OF 17 REFERENCES

### THE ZERO-DIVISOR GRAPH OF VON NEUMANN REGULAR RINGS

- 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…

### Beck′s Coloring of a Commutative Ring

- 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

- Mathematics
- 2004

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…

### The zero-divisor graph of a commutative semigroup

- 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…

### The Zero-Divisor Graph of a Commutative Ring☆

- 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).

### Lectures on Rings and Modules

- Mathematics
- 1976

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

- Mathematics
- 2002

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…