A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING

David F Anderson,Elizabeth F Lewis

doi:10.24330/ieja.266187

Abstract

Let R be a commutative ring with 1 6= 0, I a proper ideal of R, and ∼ a multiplicative congruence relation on R. Let R/∼ = { [x]∼ | x ∈ R } be the commutative monoid of ∼-congruence classes under the induced multiplication [x]∼[y]∼ = [xy]∼, and let Z(R/∼) be the set of zero-divisors of R/∼. The ∼-zero-divisor graph of R is the (simple) graph Γ∼(R) with vertices Z(R/∼) \{[0]∼} and with distinct vertices [x]∼ and [y]∼ adjacent if and only if [x]∼[y]∼ = [0]∼. Special cases include the usual zero-divisor graphs Γ(R) and Γ(R/I), the ideal-based zero-divisor graph ΓI (R), and the compressed zero-divisor graphs ΓE(R) and ΓE(R/I). In this paper, we investigate the structure and relationship between the various ∼-zero-divisor graphs.

Highlights

Introduction and definitionsLet R be a commutative ring with 1 = 0, and let Z(R) be its set of zerodivisors
We investigate the structure and relationship between the various ∼-zero-divisor graphs
The ideal-based zero-divisor graph of R with respect to an ideal I of R is the graph ΓI (R) with vertices { x ∈ R \ I | xy ∈ I for some y ∈ R \ I } and with distinct vertices x and y adjacent if and only if xy ∈ I

Summary

Introduction and definitions

The ideal-based zero-divisor graph of R with respect to an ideal I of R is the (simple) graph ΓI (R) with vertices { x ∈ R \ I | xy ∈ I for some y ∈ R \ I } and with distinct vertices x and y adjacent if and only if xy ∈ I. For any (multiplicative) commutative semigroup S with 0, let Z(S) = { x ∈ S | xy = 0 for some 0 = y ∈ S } be the set of zero-divisors of S. We introduce a unifying concept of zero-divisor graph over a commutative ring R with 1 = 0 based on a multiplicative congruence relation ∼ on R (i.e., ∼ is an equivalence relation on R and x ∼ y implies xz ∼ yz for x, y, z ∈ R). Most of the results in the first four sections of this paper are from the secondnamed author’s PhD dissertation [20] at The University of Tennessee under the direction of the first-named author

Basic results

Zero-divisor graph maps

More zero-divisor graph maps

Induced zero-divisor graph maps