## Doctoral Dissertations

#### Date of Award

5-2014

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

David F. Anderson

#### Committee Members

Michael Berry, Luis Finotti, Shashikant Mulay

#### Abstract

Let *R* be a commutative ring with nonzero identity and *I* an ideal of *R*. The focus of this research is on a generalization of the zero-divisor graph called the ideal-based zero-divisor graph for commutative rings with nonzero identity. We consider such a graph to be nontrivial when it is nonempty and distinct from the zero-divisor graph of *R*. We begin by classifying all rings which have nontrivial ideal-based zero-divisor graph complete on fewer than 5 vertices. We also classify when such graphs are complete on a prime number of vertices. In addition we classify all rings which admit nontrivial planar ideal-based zero-divisor graph. The ideas of complemented and uniquely complemented are considered for such graphs, and we classify when they are uniquely complemented. The relationship between graph isomorphisms of the ideal-based zero divisor graph with respect to *I* and graph isomorphisms of the zero-divisor graph of *R/I* [R mod I] is also considered. In the later chapters, we consider properties of ideal-based zero-divisor graphs when the corresponding factor rings are Boolean or reduced. We conclude by giving all nontrivial ideal based zero-divisor graphs on less than 8 vertices, a few miscellaneous results, and some questions for future research.

#### Recommended Citation

Smith, Jesse Gerald Jr., "Properties of Ideal-Based Zero-Divisor Graphs of Commutative Rings. " PhD diss., University of Tennessee, 2014.

https://trace.tennessee.edu/utk_graddiss/2729