Noetherian Rings Whose Annihilating-Ideal Graphs Have finite Genus

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ 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 $γ(\Bbb{AG}(R))$. It is shown that if $R$ is a Noetherian ring such that $0<γ(\Bbb{AG}(R))<\infty$, then $R$ has only finitely many ideals.

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