Rings Whose Annihilating-Ideal Graphs Have Positive Genus

TL;DR

This paper studies the properties of annihilating-ideal graphs of commutative rings, especially focusing on rings with graphs of positive genus, and characterizes certain Artinian and Gorenstein rings based on these graph properties.

Contribution

It characterizes rings with annihilating-ideal graphs of finite genus, especially Artinian and Gorenstein rings, and establishes finiteness results for classes of such rings.

Findings

Rings with finite genus annihilating-ideal graphs are either finite-ideal or Gorenstein with specific maximal ideal properties.

Finitely many isomorphism classes of Artinian rings exist with bounded genus and residue field size.

Non-domain Noetherian local rings with finite genus are either Gorenstein or Artinian with finitely many ideals.

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.