More on the Annihilator-Ideal Graph of a Commutative Ring

TL;DR

This paper explores the properties of the annihilator-ideal graph of a commutative ring, comparing it with the annihilating-ideal graph, and characterizes rings based on the graph's structure and properties.

Contribution

It introduces new characterizations of the annihilator-ideal graph, especially when it differs from the annihilating-ideal graph and has a specific girth, including conditions for it to be a star graph.

Findings

Characterization of rings where $A_I(R) eq ext{AIG}(R)$ and girth is 4.

Necessary and sufficient conditions for $A_I(R)$ to be a star graph.

Analysis of the relationship between $A_I(R)$ and $ ext{AIG}(R)$.

Abstract

Let $R$ be a commutative ring with identity and $\Bbb A (R)$ be the set of ideals of $R$ with non-zero annihilator. The annihilator-ideal graph of $R$, denoted by $A_{I} (R) $, is a simple graph with the vertex set $\Bbb A(R)^{\ast} := \Bbb A (R) \setminus\lbrace (0) \rbrace $, and two distinct vertices $I$ and $J$ are adjacent if and only if $\mathrm{Ann} _{R} (IJ) \neq \mathrm{Ann} _{R} (I) \cup \mathrm{Ann} _{R} (J)$. In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph $\Bbb A \Bbb G (R)$ (a well-known graph with the same vertices and two distinct vertices $I,J$ are adjacent if and only if $IJ=0$) associated with $R$. All rings whose $A_{I}(R) \neq \Bbb A \Bbb G (R)$ and $\mathrm{gr} (A_{I}(R)) =4$ are characterized. Among other results, we obtain necessary and sufficient conditions under which $A_{I} (R)$ is a star graph.