GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

David F Anderson,Grace Mcclurkin

doi:10.24330/ieja.663079

Abstract

Let $R$ be a commutative ring with $1 \neq 0$ and $Z(R)$ its set of zero-divisors. The zero-divisor graph of $R$ is the (simple) graph $\Gamma(R)$ with vertices $Z(R) \setminus \{0\}$, and distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. In this paper, we consider generalizations of $\Gamma(R)$ by modifying the vertices or adjacency relations of $\Gamma(R)$. In particular, we study the extended zero-divisor graph $\overline{\Gamma}(R)$, the annihilator graph $AG(R)$, and their analogs for ideal-based and congruence-based graphs.

Highlights

Let R be a commutative ring with 1 = 0, and let Z(R) be its set of zero-divisors
Starting with Γ(R), we can modify both the vertices and edges to get new “zerodivisor” graphs. We study these ideas in more detail
We show that Γ(T (R)) ∼= Γ(R) as well

Summary

Introduction

The extended zero-divisor graph of R is the (simple) graph Γ(R) with vertices Z(R)∗, and distinct vertices x and y are adjacent if and only if xmyn = 0 for positive integers m and n with xm = 0 and yn = 0. The ideal-based zero-divisor graph of R with respect to an ideal I of R is the (simple) graph ΓI (R) with vertices ZI (R) = { x ∈ R\I | xy ∈ I for some y ∈ R\I }, and distinct vertices x and y are adjacent if and only if xy ∈ I. The ideal-based extended zero-divisor graph of R (with respect to I) is the (simple) graph ΓI (R) with vertices ZI (R), and distinct vertices x and y are adjacent if and only if xmyn ∈ I for positive integers m and n with xm ∈ I and yn ∈ I. Many of the results in this paper are from the second-named author’s PhD dissertation ([28]) at The University of Tennessee under the direction of the firstnamed author

Ideal-based graphs

Congruence-based graphs

Compressed graphs

Maps between graphs