DOI

10.17077/etd.3ny5-7920

Document Type

Dissertation

Date of Degree

Spring 2019

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Anderson, Dan

First Committee Member

Bleher, Frauke

Second Committee Member

Camillo, Victor

Third Committee Member

Iovanov, Miodrag

Fourth Committee Member

Kinser, Ryan

Abstract

Let R be an integral domain. An atom is a nonzero nonunit x of R where x = yz implies that either y or z is a unit. We say that R is an atomic domain if each nonzero nonunit is a finite product of atoms. An atomic domain with only finitely many nonassociate atoms is called a Cohen-Kaplansky (CK) domain. We will investigate atoms in integral domains R with a unique maximal ideal M. Of particular interest will be atoms that are not in M^2.

After studying the atoms in integral domains, we will narrow our focus to CK domains with a unique maximal ideal M. In this pursuit, we investigate atoms in M^2 for these CK domains. We will show that the minimal number of atoms needed to have an atom in M^2 is exactly eight. This disproves a conjecture given by Cohen and Kaplansky in 1946 that the minimal number would be ten. We then classify complete local CK domains with exactly three atoms.

Keywords

Atoms, CK, CK-Domain, Domains, Integral, Quasilocal

Pages

vii, 72 pages

Bibliography

Includes bibliographical references (page 72).

Copyright

Copyright © 2019 Kevin Wilson Bombardier

Included in

Mathematics Commons

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