On Perfectness of Annihilating-Ideal Graph of $\mathbb{Z}_n$

TL;DR

This paper characterizes the integers n for which the annihilating-ideal graph of the ring Z_n is perfect, extending previous work that showed it is weakly perfect.

Contribution

It provides a complete characterization of n such that the annihilating-ideal graph of Z_n is perfect, refining earlier results on its weak perfection.

Findings

Identifies conditions on n for which the graph is perfect.

Complements prior work on weak perfection of the graph.

Enhances understanding of the structure of annihilating-ideal graphs.

Abstract

The annihilating-ideal graph of a commutative ring $R$ with unity is defined as the graph $\mathbb{AG}(R)$ with the vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ = 0$. Nikandish {\it et.al.} proved that $\mathbb{AG}(\mathbb{Z}_n)$ is weakly perfect. In this short paper, we characterize $n$ for which $\mathbb{AG}(\mathbb{Z}_n)$ is perfect.