on the genus of annihilator intersection graph of commutative rings

Let r be a commutative ring with unity and a(r) be the set of annihilatingideals of r. the annihilator intersection graph of r, represented by aig(r), is an undirected graph with a(r) ∗ as the vertex set and m ∼ n is an edge of aig(r) if and only if ann(mn) ̸= ann(m) ∩ ann(n), for distinct vertices m and n of aig(r). in this paper, we first defined finite commutative rings whose annihilator intersection graph is isomorphic to various well-known graphs, and then all finite commutative rings with a planar or toroidal annihilator intersection graph were characterized.