On the genus of the zero-divisor graph of \(\mathbb Z_n\) (Q471791)

Summary: Let \(R\) be a commutative ring with identity. The zero-divisor graph of \(R\), denoted \(\Gamma(R)\), is the simple graph whose vertices are the nonzero zero-divisors of \(R\), and two distinct vertices \(x\) and \(y\) are linked by an edge if and only if \(xy=0\). The genus of a simple graph \(G\) is the smallest integer \(g\) such that \(G\) can be embedded into an orientable surface \(S_g\). In this paper, we determine that the genus of the zero-divisor graph of \(\mathbb Z_n\), the ring of integers modulo \(n\), is two or three.