classification of rings with toroidal annihilating-ideal graph

Let r be a non-domain commutative ring with identity and a∗(r) be the set of non-zero ideals with non-zero annihilators. we call an ideal i₁ of r, an annihilating-ideal if there exists a non-zero ideal i₂ of r such that i₁i₂ = (0). the annihilating-ideal graph of r is defined as the graph ag(r) with the vertex set a∗(r) and two distinct vertices i₁ and i₂ are adjacent if and only if i₁i₂ = (0). in this paper, we characterize all commutative artinian non-local rings r for which ag(r) has genus ne.