Date of Award

5-2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Member

Dr. James Coykendall, Committee Chair

Committee Member

Dr. Sean Sather-Wagstaff

Committee Member

Dr. Kevin James

Committee Member

Dr. Matthew Macauley

Abstract

An integral domain D is atomic if every non-zero non-unit is a product of irreducibles. More generally, D is quasi-atomic if every non-zero non-unit divides some product of atoms. Arbitrary integral domains, however, cannot be assumed to be quasi-atomic in general; factorization in a non-atomic D can be subtle. We outline a novel method of qualifying the quasi-atomicity of D by studying ascending filtrations of localizations of D and the associated groups of divisibility. This approach yields structure theorems, cochain complexes, and cohomological results. We take care to present examples of integral domains exhibiting the spectrum of factorization behavior and we relate the results of our new method to factorization in D.

Included in

Mathematics Commons

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