Commutative rings with toroidal zero-divisor graphs

TL;DR

This paper classifies finite commutative rings based on whether their zero-divisor graphs can be embedded on a torus or plane, focusing on genus number and topological properties.

Contribution

It characterizes all finite commutative rings whose zero-divisor graphs are either toroidal or planar, providing a complete classification.

Findings

Identifies rings with zero-divisor graphs embeddable on a torus or plane.

Provides a classification of finite commutative rings based on graph genus.

Illustrates the structure of rings with specific topological graph properties.

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.