## Doctoral Dissertations

#### Date of Award

8-2015

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

David F. Anderson

#### Committee Members

Shashikant B. Mulay, Marie K. Jameson, Donald J. Bruce

#### Abstract

Let *R* be a commutative ring with nonzero identity and ~ a multiplicative congruence relation on *R*. Then, *R*/~ is a semigroup with multiplication [*x*][*y*] = [*xy*], where [*x*] is the congruence class of an element *x* of *R*. We define the congruence-based zero-divisor graph of R ito be the simple graph with vertices the nonzero zero-divisors of *R*/~ and with an edge between distinct vertices [*x*] and [*y*] if and only if [*x*][*y*] = [0]. Examples include the usual zero-divisor graph of *R*, compressed zero-divisor graph of *R*, and ideal-based zero-divisor graph of *R*. We study relationships among congruence-based zero-divisor graphs for various congruence relations on *R*. In particular, we study connections between ring-theoretic properties of *R* and graph-theoretic properties of congruence-based zero-divisor graphs for various congruence relations on *R*.

#### Recommended Citation

Lewis, Elizabeth Fowler, "The Congruence-Based Zero-Divisor Graph. " PhD diss., University of Tennessee, 2015.

https://trace.tennessee.edu/utk_graddiss/3436