Date

2012

Department or Program

Mathematics

Primary Wellesley Thesis Advisor

Alexander Diesl

Abstract

A commutative ring R can be represented as a graph whose vertices are the ideals of R, and in which two vertices v1; v2 are adjacent if and only if v1 and v2 are comaximal. This graph, denoted G(R); is called the comaximal ideal graph of R and is a variant on the graphical ring representations of Beck and of Sharma and Bhatwadekar. Because the properties of G(R) are derived from the lattice of ideals, I have been able to use this ring representation to highlight overall structural properties not visible with the other element based representations. Using lattice theory I have shown that there are specific relations with the clique number, chromatic number and number of partitions of a graph, and that there is a correspondence between the vertices not contained in the Jacobson radical J(R) and those contained in J(R).

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