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

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.