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

Rings having zero-divisor graphs of
small diameter or large girth

S.B. Mulay

Received: 30th August, 2005



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


  1. D.F. Anderson and P.S. Livingston;
    The zero-divisor graph of a commutative ring,
    J. Algebra 217 (1999), pp. 434--447. MR1700509
  2. 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
  3. I. Beck;
    Coloring of commutative rings,
    J. Algebra 116 (1988), pp. 208--226. MR944156
  4. S.B. Mulay;
    Cycles and symmetries of zero-divisors,
    Comm, Algebra 30 (2002), pp. 3533--3558. MR1915011
  5. M. Nagata;
    Local rings (Krieger Publishing Company, Huntington, N.Y., 1975). MR460307

ISSN 0004-9727