Infinite Rings with Planar Zero-Divisor Graphs

For any commutative ring R that is not a domain, there is a zero-divisor graph, denoted (R), in which the vertices are the nonzero zero-divisors of R and two distinct vertices x and y are joined by an edge exactly when xy=0. Smith (2007) characterized the graph structure of (R) provided it is infinite and planar. In this article, we give a ring-theoretic characterization of R such that (R) is infinite and planar.