Dissertation/Thesis Abstract

Assessing Non-Atomicity in Groups of Divisibility
by Goodell, Brandon G., Ph.D., Clemson University, 2017, 97; 10275474
Abstract (Summary)

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.

Indexing (document details)
Advisor: Coykendall, James
Commitee: James, Kevin, Macauley, Matthew, Sather-Wagstaff, Sean
School: Clemson University
Department: Mathematical Science
School Location: United States -- South Carolina
Source: DAI-B 78/11(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Factorization, Integral domains, Partially ordered groups
Publication Number: 10275474
ISBN: 9780355026740
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