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Full-Text Articles in Mathematics
Various Topics On Graphical Structures Placed On Commutative Rings, Darrin Weber
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
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.
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 …