Document Type


Date of Degree

Spring 2010

Degree Name

PhD (Doctor of Philosophy)

Degree In


First Advisor

Anderson, Dan D

First Committee Member

Bleher, Frauke

Second Committee Member

Camillo, Vic

Third Committee Member

Dykstra, Richard

Fourth Committee Member

Muhly, Paul


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).


Abstract Factorization, Commutative Rings


iv, 67 pages


Includes bibliographical references (pages 66-67).


Copyright 2010 Jeremiah N Reinkoester

Included in

Mathematics Commons