The diameter of a zero divisor graph

Abstract

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.