On the Genus of the Zero-Divisor Graph of Zn

Abstract

Let be a commutative ring with identity. The zero-divisor graph of , denoted Γ( ), is the simple graph whose vertices are the nonzero zero-divisors of , and two distinct vertices and are linked by an edge if and only if = 0. The genus of a simple graph is the smallest integer such that can be embedded into an orientable surface . In this paper, we determine that the genus of the zero-divisor graph of Z , the ring of integers modulo , is two or three.