Properties of Ideal-Based Zero-Divisor Graphs of Commutative Rings

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Smith, Jesse Gerald, Jr.
zero-divisor; graph; planar; Algebra
thesis / dissertation description
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.