Bulletin of the Australian Mathematical
Society
Bull. Austral. Math. Soc.
0004-9727
ALNBAB
2005
72
3
10.wxyz/CV72P3
http://www.austms.org.au/Publ/Bulletin/V72P3/
Rings having zero-divisor graphs of small diameter or
large girth
S.B.
Mulay
13 February 2006
2006
2
13
481
490
723-5248-Mulay-2005
10.wxyz/C2005V72P3p481
http://www.austms.org.au/Publ/Bulletin/V72P3/723-5248-Mulay/
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 $3$.
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