#### Date of Award

5-2001

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

D. F. Anderson

#### Committee Members

S. B. Mulay, David E. Dobbs, Michael W. Berry

#### Abstract

Let R be a commutative ring with 1, and let *Z*(*R*) denote the set of zerodivisors of R. We define an undirected graph Γ(*R*) with vertices *Z*(*R*)* = *Z*(*R*) - {0}, where distinct vertices *x* and *y* of R are connected if and only if *xy* = 0. This graph is called the zero-divisor graph of R. We extend the definition of the zero-divisor graph to a noncommutative ring in several ways. Next, given a commutative ring R and ideal *Ι* of R, we introduce the notion of an ideal-based graph. This is an undirected graph with vertex set {x ∈ *R* – *Ι*| *xy* ∈ *Ι* *for some* y ∈ *R* - *Ι*}, where distinct vertices *x* and *y* are adjacent if and only if *xy* ∈ *Ι*. The properties of such a graph are investigated. We give several results concerning the zero-divisor graph of a commutative ring. Finally, the appendix gives examples illustrating an equivalence relation on the vertices of Γ(*R*) that can be used to produce a related graph for rings R of specific types.

#### Recommended Citation

Redmond, Shane Patrick, "Generalizations of the Zero-Divisor Graph of a Ring. " PhD diss., University of Tennessee, 2001.

http://trace.tennessee.edu/utk_graddiss/1504