It is shown that, for a fixed positive integer g, there are finitely many isomorphism classes of rings whose zero-divisor graph has genus g. The proof can then be modified to yield an analogous result for nonorientable genus.Throughout this note, all rings are assumed to be commutative rings with identity. We denote the ring of integers modulo n by Z n and the field with q elements by F q . If S is a subset of a ring R, then S * will denote the set of nonzero elements of S. An element a ∈ R is said to be a zero divisor of R if there is an element b of R * such that ab = 0. The set of zero divisors of R is denoted Z (R). The zerodivisor graph of R, denoted Γ (R), is the (simple) graph whose vertex set is Z (R) * and whose edge set is E = {{a, b} | a, b ∈ Z (R) * and ab = 0}. This definition of Γ (R) is the same as that introduced in [4]. It is clear that Γ (R) is empty if and only if R has no zero divisors, i.e., R is an integral domain.Example 1. The vertex set for Γ (Z 6 ) is {2, 3, 4} and the zero divisor graph is shown in Fig. 1.The vertex set for Γ (Z 16 ) is {2, 4, 6, 8, 10, 12, 14} and the zero divisor graph is shown in Fig. 2.Example 2. Let R = Z 8 [X]/(2X, X 2 ), and let x denote the image of X in R. Then the vertex set for Γ (R) is the set (2, x) * = {2, 4, 6, x, 2 + x, 4 + x, 6 + x} of nonzero elements of the maximal ideal, and the zero divisor graph is shown in Fig. 3.
