Description of finite nonnilpotent rings with planar zero-divisor graphs

The zero-divisor graph of an associative ring $R$ is a graph whose vertices are all nonzero (one-sided and two-sided) zero divisors of $R$, two distinct vertices $x,y$ are connected by an edge if and only if $xy=0$ or $yx=0$.
In this paper, all finite nonnilpotent rings with planar zero-divisor graphs are completely described. In the previous paper by Kuzmina and Maltsev, the finite nilpotent rings with planar zero-divisor graphs were studied. Thus, this paper completes the description of finite rings with planar zero-divisor graphs.