Full Descripion of ring varieties whose finite rings are uniquely determined by their zero-divisor graphs

TL;DR

This paper characterizes specific ring varieties where the zero-divisor graph uniquely determines all finite rings within those varieties, providing a complete classification.

Contribution

It offers a comprehensive description of ring varieties with the property that finite rings are uniquely identified by their zero-divisor graphs.

Findings

Complete classification of such ring varieties

Identification of conditions for zero-divisor graph uniqueness

Extension of zero-divisor graph theory to ring varieties

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 give a full description of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.