Rings Whose Annihilating-Ideal Graphs Have Positive Genus

Abstract

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 the vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We investigate commutative rings $R$ whose annihilating-ideal graphs have positive genus $\gamma(\Bbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $\gamma(\Bbb{AG}(R))<\infty$, then $R$ has finitely many ideals or $(R,\mathfrak{m})$ is a Gorenstein ring with maximal ideal $\mathfrak{m}$ and ${\rm v.dim}_{R/{\mathfrak{m}}}{\mathfrak{m}}/{\mathfrak{m}}^{2}=2$. Also, for any two integers $g\geq 0$ and $q>0$, there are finitely many isomorphism classes of Artinian rings $R$ satisfying the conditions: (i) $\gamma(\Bbb{AG}(R)) < g$ and (ii) $|R/{\mathfrak{m}}| \leq q$ for every maximal ideal ${\mathfrak{m}}$ of $R$. Also, it is shown that if $R$ is a non-domain Noetherian local ring such that $\gamma(\Bbb{AG}(R))<\infty$, then either $R$ is a Gorenstein ring or $R$ is an Artinian ring with finitely many ideals.