Zero-Annihilator Graphs of Commutative Rings

TL;DR

This paper introduces the zero-annihilator graph of a commutative ring, exploring its properties based on the intersections of annihilator ideals of nonzero, nonunit elements.

Contribution

It defines and investigates the properties of the zero-annihilator graph for commutative rings, a new concept linking ring theory and graph theory.

Findings

Characterization of adjacency in zero-annihilator graphs

Connections between ring properties and graph structure

Potential applications in algebraic graph theory

Abstract

Assume that $R$ is a commutative ring with nonzero identity. In this paper, we introduce and investigate zero-annihilator graph of $R$ denoted by $\mathtt{ZA}(R)$. It is the graph whose vertex set is the set of all nonzero nonunit elements of $R$ and two distinct vertices $x$ and $y$ are adjacent whenever ${\rm Ann}_R(x)\cap {\rm Ann}_R(y)=\{0\}$.