Let be a commutative ring with 1 and let be its set of nonzero zero divisors. The zero divisor graph of has vertices with two vertices and connected by an edge if and only if . D. F. Anderson and P. S. Livingston [J. Algebra 217, 434–447 (1999; Zbl 0941.05062)] showed that is connected, has diameter , and characterized the rings with . M. Axtell, J. Coykendall and J. Stickles [Commun. Algebra 33, 2043–2050 (2005; Zbl 1088.13006)] investigated the zero divisor graphs of and and showed among other things that for noetherian not isomorphic to , if one of , , or has diameter 2, then so do the other two.
The main result of the paper under review is that for a reduced ring that is not an integral domain, we have and all possible sequences for these three diameters are given. For example, if and only if has more than two minimal primes and there is a pair of zero divisors and such that does not have nonzero annihilator. A similar characterization is given for and when is not reduced.