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
Recommended Citation
Edmonds, Ranthony A.C.. "Factorization in polynomial rings with zero divisors." PhD (Doctor of Philosophy) thesis, University of Iowa, 2018.
https://doi.org/10.17077/etd.1r0asgxa