Properties of IdealBased ZeroDivisor Graphs of Commutative Rings
 Publication Year:
 2014

 Bepress 433

 Bepress 113
 Repository URL:
 https://trace.tennessee.edu/utk_graddiss/2729
 Author(s):
 Tags:
 zerodivisor; 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 zerodivisor graph called the idealbased zerodivisor graph for commutative rings with nonzero identity. We consider such a graph to be nontrivial when it is nonempty and distinct from the zerodivisor graph of R. We begin by classifying all rings which have nontrivial idealbased zerodivisor 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 idealbased zerodivisor 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 idealbased zero divisor graph with respect to I and graph isomorphisms of the zerodivisor graph of R/I [R mod I] is also considered. In the later chapters, we consider properties of idealbased zerodivisor graphs when the corresponding factor rings are Boolean or reduced. We conclude by giving all nontrivial ideal based zerodivisor graphs on less than 8 vertices, a few miscellaneous results, and some questions for future research.