When the annihilator graph of a commutative ring is planar or toroidal?

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) not equal ann (R)(x) boolean OR ann (R)(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.