Zero Divisor Graphs and Poset Decomposition

<p>A graph is associated to any commutative ring R where the vertices are the non-zero zero divisors of R with two vertices adjacent if x · y = 0. The zero-divisor graph has also been studied for various algebraic stuctures such as semigroups and partially ordered sets. In this paper, we will discuss some known results on zero-divisor graphs of posets as well as the concept of compactness as it relates to zero-divisor graphs. We will dicuss equivalence class graphs defined on the elements of various algebraic structures and also the reduced graph defined on the vertices of a compact graph. After introducing and discussing some known results on poset dimension, we will show that poset decomposition can be directly related to the equivalence classes represented in a reduced graph. Using this decomposition, we can build a poset of a compact graph with any dimension in a specified interval. Thus we have a device which gives us the ability to study the dimension of a poset of a zero-divisor graph.</p>

Name thesis:degree_name

M.S. in Mathematics

Level thesis:degree_level

Thesis

Year dc:date.available

2011

Author dc:creator

• Putnam, Bette Catherine

Contributors dc:contributor

• Laura Sheppardson
• William Staton
• Sandra Spiroff

• Poset Dimension
• Zero-Divisor
• Zero-Divisor Graphs
• Mathematics

Repository record dc:identifier

https://egrove.olemiss.edu/etd/237

OAI identifier oai:identifier

oai:egrove.olemiss.edu:etd-1236