When the Annihilator Graph of a Commutative Ring Is Planar or Toroidal?

Abstract

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the undirected graph AG(R) with the vertex set Z(R) * = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if ann R (xy) = ann R (x) ∪ ann R (y). In this paper, all rings whose annihilator graphs can be embed on the plane or torus are classified.