On the diameter and girth of a zero-divisor graph

For a commutative ring R with zero-divisors Z(R), the zero-divisor graph of R is Gamma(R) = Z(R) - {0}, with distinct vertices x and y adjacent if and only if xy = 0. In this paper, we characterize when either diam(Gamma(R)) <= 2 or gr(Gamma(R)) >= 4. We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations. (C) 2006 Elsevier B.V. All rights reserved.