The diameter of a zero divisor graph

Let R be a commutative ring and let Z∗(R) be its set of nonzero zero divisors. The set Z∗(R) makes up the vertices of the corresponding zero divisor graph, Γ(R), with two distinct vertices forming an edge if the product of the two elements is zero. The distance between vertices a and b (not necessarily distinct from a) is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ(R)), is the sup of these distances. For a reduced ring R with nonzero zero divisors, 1⩽diam(Γ(R))⩽diam(Γ(R[x]))⩽diam(Γ(R〚x〛))⩽3. A complete characterization for the possible diameters is given exclusively in terms of the ideals of R. A similar characterization is given for diam(Γ(R)) and diam(Γ(R[x])) when R is nonreduced. Various examples are provided to illustrate the difficulty in dealing with the power series ring over a nonreduced ring.