DOI

10.17077/etd.1r0asgxa

Document Type

Dissertation

Date of Degree

Summer 2018

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Anderson, Daniel D.

First Committee Member

Kutzko, Philip

Second Committee Member

Krishnamurthy, Muthu

Third Committee Member

Camillo, Victor

Fourth Committee Member

Kinser, Ryan

Abstract

Factorization theory is concerned with the decomposition of mathematical objects. Such an object could be a polynomial, a number in the set of integers, or more generally an element in a ring. A classic example of a ring is the set of integers. If we take any two integers, for example 2 and 3, we know that $2 \cdot 3=3\cdot 2$, which shows that multiplication is commutative. Thus, the integers are a commutative ring. Also, if we take any two integers, call them $a$ and $b$, and their product $a\cdot b=0$, we know that $a$ or $b$ must be $0$. Any ring that possesses this property is called an integral domain. If there exist two nonzero elements, however, whose product is zero we call such elements zero divisors. This thesis focuses on factorization in commutative rings with zero divisors.

In this work we extend the theory of factorization in commutative rings to polynomial rings with zero divisors. For a commutative ring $R$ with identity and its polynomial extension $R[X]$ the following questions are considered: if one of these rings has a certain factorization property, does the other? If not, what conditions must be in place for the answer to be yes? If there are no suitable conditions, are there counterexamples that demonstrate a polynomial ring can possess one factorization property and not another? Examples are given with respect to the properties of atomicity and ACCP. The central result is a comprehensive characterization of when $R[X]$ is a unique factorization ring.

Keywords

commutative ring theory, factorization, polynomial rings, zero divisors

Pages

v, 80 pages

Bibliography

Includes bibliographical references (pages 79-80).

Copyright

Copyright © 2018 Ranthony A.C. Edmonds

Included in

Mathematics Commons

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