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 zerodivisor graphs were studied. Thus, this paper completes the description of finite rings with planar zero-divisor graphs.