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 Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we characterize when either image or image. We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.