On varieties of rings whose finite rings are determined by their zero-divisor graphs

TL;DR

This paper investigates specific ring varieties where the structure of finite rings can be uniquely identified solely through their zero-divisor graphs, enhancing understanding of the relationship between ring properties and graph representations.

Contribution

It introduces conditions under which finite rings are uniquely determined by their zero-divisor graphs within certain ring varieties.

Findings

Finite rings are uniquely determined by their zero-divisor graphs in specific varieties.

Properties of zero-divisor graphs reflect underlying ring structures.

New characterizations of ring varieties based on zero-divisor graph uniqueness.

Abstract

The zero-divisor graph $\Gamma(R)$ of an associative ring $R$ is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of $R$, and two distinct vertices $x$ and $y$ are joined by an edge iff either $xy=0$ or $yx=0$. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.