Mathematics Commons^™

Various Topics On Graphical Structures Placed On Commutative Rings, Darrin Weber

Doctoral Dissertations

In this dissertation, we look at two types of graphs that can be placed on a commutative ring: the zero-divisor graph and the ideal-based zero-divisor graph. A zero-divisor graph is a graph whose vertices are the nonzero zero-divisors of a ring and two vertices are connected by an edge if and only if their product is 0. We classify, up to isomorphism, all commutative rings without identity that have a zero-divisor graph on 14 or fewer vertices.

An ideal-based zero-divisor graph is a generalization of the zero-divisor graph where for a ring R and ideal I the vertices are { …

The Congruence-Based Zero-Divisor Graph, Elizabeth Fowler Lewis

Doctoral Dissertations

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 …

Properties Of Ideal-Based Zero-Divisor Graphs Of Commutative Rings, Jesse Gerald Smith Jr.

Doctoral Dissertations

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 …