#### Title

#### Document Type

Dissertation

#### Date of Degree

Spring 2010

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Dan D. Anderson

#### 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 = λa _{1} · · · a_{n}* where

*λ*is in

*U (D)*and

*a*is

_{i}*τ*-related to

*a*, denoted

_{j}*a*, for

_{i}τ a_{j}*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)*. We then write

^{∗}= D*a τ*(resp.

_{max}b*a τ*, or

_{v}b, a τ_{[ ]}b*a τ*) if

_{∗}b*a*and

*b*are comaximal (resp. v -coprime, coprime, or ∗-coprime).

#### Keywords

Abstract Factorization, Commutative Rings

#### Pages

iv, 67

#### Bibliography

66-67

#### Copyright

Copyright 2010 Jeremiah N Reinkoester