Document Type

Dissertation

Date of Degree

Spring 2010

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Dan D. Anderson

First Committee Member

Frauke Bleher

Second Committee Member

Vic Camillo

Third Committee Member

Richard Dykstra

Fourth Committee Member

Paul Muhly

Abstract

In [2], Dan Anderson and Andrea Frazier introduced a generalized theory of factorization. Given a relation τ on the nonzero, nonunit elements of an integral domain D, they defined a τ-factorization of a to be any proper factorization a = λa1 · · · an where λ is in U (D) and ai is τ-related to aj, denoted ai τ aj, for i not equal to j . From here they developed an abstract theory of factorization that generalized factorization in the usual sense. They were able to develop a number of results analogous to results already known for usual factorization.

Our work focuses on the notion of τ-factorization when the relation τ has characteristics similar to those of coprimeness. We seek to characterize such τ-factorizations. For example, let D be an integral domain with nonzero, nonunit elements a, b ∈ D. We say that a and b are comaximal (resp. v-coprime, coprime ) if (a, b) = D (resp., (a, b)v = D, [a, b] = 1). More generally, if ∗ is a star-operation on D, a and b are ∗-coprime if (a, b) = D. We then write a τmax b (resp. a τv b, a τ[ ] b, or a τ b) if a and b are comaximal (resp. v -coprime, coprime, or ∗-coprime).

Keywords

Abstract Factorization, Commutative Rings

Pages

iv, 67 pages

Bibliography

Includes bibliographical references (pages 66-67).

Copyright

Copyright 2010 Jeremiah N Reinkoester

Included in

Mathematics Commons

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