Characterization of rings with genus two cozero-divisor graphs

TL;DR

This paper characterizes finite non-local commutative rings whose cozero-divisor and reduced cozero-divisor graphs have genus two, linking algebraic properties of rings with topological graph features.

Contribution

It provides a complete classification of such rings based on the genus of their associated cozero-divisor graphs.

Findings

Identifies all finite non-local commutative rings with genus two cozero-divisor graphs.

Characterizes the structure of rings via topological properties of their graphs.

Links algebraic ring properties to graph genus in a comprehensive manner.

Abstract

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$ is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x \notin Ry$ and $y \notin Rx$. The reduced cozero-divisor graph of a ring $R$, is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of $R$ and two distinct vertices $(a)$ and $(b)$ are adjacent if and only if $(a) \not\subset (b)$ and $(b) \not\subset (a)$. In this paper, we characterize all classes of finite non-local commutative rings for which the cozero-divisor graph and reduced cozero-divisor graph is of genus two.