Commutative rings with toroidal zero-divisor graphs

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.