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

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.