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.