Rings Whose Annihilating-Ideal Graphs Have Positive Genus

Abstract

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 $\mathbb{AG}(R)$ with the vertex set $\mathbb{A}(R{)}^{\ast }=\mathbb{A}\setminus \{(0)\}$ 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 $\gamma (\mathbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $\gamma (\mathbb{AG}(R))<\infty$, then $R$ has finitely many ideals or $(R,\mathfrak{m})$ is a Gorenstein ring with maximal ideal $\mathfrak{m}$ and ${v.dim}_{R/\mathfrak{m}}\mathfrak{m}/{\mathfrak{m}}^2=2$. Also, for any two integers $g\ge 0$ and $q>0$, there are finitely many isomorphism classes of Artinian rings $R$ satisfying the conditions: (i) $\gamma (\mathbb{AG}(R))<g$ and (ii) $|R/\mathfrak{m}|\le q$ for every maximal ideal $\mathfrak{m}$ of $R$. Also, it is shown that if $R$ is a non-domain Noetherian local ring such that $\gamma (\mathbb{AG}(R))<\infty$, then either $R$ is a Gorenstein ring or $R$ is an Artinian ring with finitely many ideals.

13 pages