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.
Categories: math.AC math.RA