*Bull. Austral. Math. Soc.* 72(3) pp.481--490, 2005.

# Rings having zero-divisor graphs of

small diameter or large girth

## S.B. Mulay |

.

## Abstract

Let

*R*be a commutative ring possessing (non-zero) zero-divisors. There is a natural graph associated to the set of zero-divisors of*R*. In this article we present a characterisation of two types of*R*. Those for which the associated zero-divisor graph has diameter different from 3 and those*R*for which the associated zero-divisor graph has girth other than 3. Thus, in a sense, for a generic non-domain*R*the associated zero-divisor graph has diameter 3 as well as girth 3.#### Click to download PDF of this article (free access until July 2006)

#### or get the *no-frills* version

[an error occurred while processing this directive]
(Metadata: XML, RSS, BibTeX) | MathSciNet: MR2199650 | Z'blatt-MATH: 1097.13007 |

## References

- D.F. Anderson and P.S. Livingston;

The zero-divisor graph of a commutative ring,

*J. Algebra***217**(1999), pp. 434--447.**MR1700509** - D.F. Anderson, R. Levy and J. Shapiro;

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

*J. Pure Appl. Algebra***180**(2003), pp. 221--241.**MR1966657** - I. Beck;

Coloring of commutative rings,

*J. Algebra***116**(1988), pp. 208--226.**MR944156** - S.B. Mulay;

Cycles and symmetries of zero-divisors,

*Comm, Algebra***30**(2002), pp. 3533--3558.**MR1915011** - M. Nagata;

*Local rings*(Krieger Publishing Company, Huntington, N.Y., 1975).**MR460307**

ISSN 0004-9727