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

Keywords:
Annihilator graph, planarity, toroidality

### 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) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.