# 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} }

## 203 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).

### 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…

### DIVISIBILITY CONDITIONS IN COMMUTATIVE RINGS WITH ZERODIVISORS

- 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…