Noetherian Rings Whose Annihilating-Ideal Graphs Have finite Genus

Let $R$ be a commutative ring and $\mathbb{A}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph $\mathbb{A}\mathbb{G}(R)$ with vertex set $\mathbb{A}(R{)}^{\ast }=\mathbb{A}\setminus \{(0)\}$ such that two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We characterize commutative Noetherian rings $R$ whose annihilating-ideal graphs have finite genus $\gamma (\mathbb{A}\mathbb{G}(R))$. It is shown that if $R$ is a Noetherian ring such that $0<\gamma (\mathbb{A}\mathbb{G}(R))<\mathrm{\infty }$, then $R$ has only finitely many ideals.