Noetherian Rings Whose Annihilating-Ideal Graphs Have finite Genus

TL;DR

This paper characterizes commutative Noetherian rings based on the finite genus of their annihilating-ideal graphs, revealing that such rings with finite genus have only finitely many ideals.

Contribution

It provides a complete characterization of Noetherian rings with annihilating-ideal graphs of finite genus, linking graph-theoretic properties to ring structure.

Findings

Rings with finite genus have finitely many ideals.

Characterization of rings based on the genus of their annihilating-ideal graphs.

Finite genus implies a finite ideal structure in Noetherian rings.

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 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 $\gamma(\Bbb{AG}(R))$. It is shown that if $R$ is a Noetherian ring such that $0<\gamma(\Bbb{AG}(R))<\infty$, then $R$ has only finitely many ideals.