Commutative rings with toroidal zero-divisor graphs

Abstract

Let \(R\) be a commutative ring and \(\Gamma(R)\) denote its zero-divisor graph. In this paper, we investigate the genus number of the compact Riemann surface which \(\Gamma(R)\) can be embedded and illustrate all finite commutative rings \(R\) (up to isomorphism) such that \(\Gamma(R)\) is either toroidal or planar.