Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2015 Restoring Factorization in Integral Domains Susan Kirk Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd © The Author Downloaded from https://scholarscompass.vcu.edu/etd/3850 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. Copyright c 2015 by Susan Kirk All rights reserved Restoring Factorization in Integral Domains A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. by Susan L. Kirk Master of Science Director: Dr. Taylor, Advisor, Associate Professor Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, Virginia April 2015 iii Acknowledgements Thanks to my family. iv Table of Contents Acknowledgements iii Abstract v 1 Preliminaries 2 1.1 Ring Theory . .2 1.2 Integral Domains . .4 2 Factorization in Integral Domains 14 2.1 General Factorization . 14 2.2 Quadratic Integer Rings . 16 Bibliography 34 Vita 34 Abstract RESTORING FACTORIZATION IN INTEGRAL DOMAINS By Susan L. Kirk, Master of Science in Mathematical Sciences A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. Virginia Commonwealth University, 2015 Director: Dr. Taylor, Advisor, Associate Professor, Department of Mathematics and Ap- plied Mathematics This is an expository thesis on integral domains which are not unique factorization do- mains. We focus on restoring a type of unique factorization using prime ideals within quadratic integer rings. In particular, we examine which quadratic integer rings will admit such factorization. In this paper we consider integral domains which lack unique factorization, such as the prime factorization that exists in the integers. We discuss a way to restore a type of unique factorization using the construct of ideals. The first chapter of our paper provides background Ring Theory and assumes basic abstract algebra knowledge. Necessary theorems are presented, some without proof, and examples are given. The second chapter focuses mainly on Quadratic Integer Rings. In particular, quadratic integer rings in which we can restore a type of factorization using prime ideals. Inter- esting history relative to the study is included where appropriate. 1 Chapter 1 Preliminaries In this chapter we provide the necessary definitions and background information from ring theory to understand the paper. Basic abstract algebra knowledge is assumed and most theorems will be stated without proof. Where appropriate, references are given. 1.1 Ring Theory Definition 1 A Ring is a non-empty set R with two binary operations: (+ addition and × multiplication) that satisfy the following properties for all a, b, c 2 R: 1. If a, b 2 R, then a + b 2 R (closure under addition) 2. a + (b + c) = (a + b) + c (associative property for addition) 3. a + b = b + a (commutative property for addition) 4. There exists a unique element 0 2 R such that 0 + a = a = a + 0 for all a 2 R (additive identity exists) 5. For every element a 2 R, there exists a unique element a-1 such that a-1+a = 0 = a+a-1 (additive inverses exist) 2 6. If a, b 2 R, then ab 2 R (closure under multiplication) 7. a(bc) = (ab)c (associative property for multiplication) 8. a(b + c) = ab + ac and (a + b)c = ac + bc (distributive properties) We denote the set R with the two binary operations as (R, +, ×). The first five properties above show that every ring is an Abelian Group under addition. Example 1 The integers Z is a ring where 0 is the additive identity and the additive identity of each element z is just -z. Example 2 The set Mn(R) of n × n matrices with entries from the ring R is a ring. The n × n zero matrix is the unique additive identiy element of Mn(R). We say a ring R is commutative if ab = ba for all a, b in R. The ring R is called a ring with unity (or a ring with identity) if R contains a multiplicative identity 1 6= 0 such that 1a = a = a1 for all a 2 R. Example 3 The set E of even integers is a commutative ring since for all a, b 2 E, ab = ba. Note that there is no even integer x such that xa = a = ax for all nonzero a 2 E, so E is not a ring with unity. Example 4 The set Mn(R) of n × n matrices with real number entries is a ring. The unique identity element of Mn(R) is the n×n identity matrix, thus Mn(R) is a ring with unity. Notice Mn(R) is not commutative since matrix multiplication is not commutative. Rings can have a several extra multiplication properties such as a multiplicative identity 1 6= 0 and multiplication can be commutative. Example 5 The set Z=6Z with addition and multiplication defined as modulo 6 is a commutative ring with unity. In fact, for any n 2 Z, where n is not prime, the quotient group Z=nZ is a commutative ring with unity. 3 A subring S of a ring R is analogous to a subgroup of a group. That is, if S is a subset of a ring R, such that: 1. if a, b 2 S, then a + b 2 S (closed under ring addition) 2. if a, b 2 S, then ab 2 S (closed under ring multiplication) 3. 0 2 S (the additive identity of the ring R is also in S) 4. if a 2 S, then there exists some a-1 2 S such that a + a-1 = 0 then S is a subring of R. Example 6 The integers Z is a subring of the rationals Q. An element a of a ring R is called a zero divisor if a 6= 0 and there exists some nonzero element b 2 R such that either ab = 0 or ba = 0. Example 7 In Z=6Z = f0, 1, 2, 3, 4, 5g, we see that 2 × 3 = 6 = 0 mod 6, but 2 6= 0 and 3 6= 0. So 2 and 3 are called zero divisors in Z=6Z. In fact, every nonzero element in Z=nZ that is not relatively prime to n is a zero divisor. 1.2 Integral Domains The type of rings that will be of focus in this paper are commutative rings with identity that have no zero divisors. Definition 2 An Integral Domain is a commutative ring R with identity 1 6= 0 that satisfies the axiom: whenever a, b 2 R and ab = 0, then a = 0 or b = 0. The condition 1 6= 0 is needed to exclude the zero ring. Notice that the axiom says that R has no zero divisors. Example 8 The set Z is an integral domain. 4 p p Example 9 Consider the set Z -17 = a + b -17 j a, b 2 Z where addition and multliplication of elements is defined as: p p p (a + b -17) + (c + d -17) = (a + c) + (b + d) -17 p p p (a + b -17)(c + d -17) = ac + (ad + bc) -17 - 17bd. p Z -17 is an integral domain. A nonzero element u in an integral domain R is called a unit if there exists some nonzero v 2 R such that uv = 1 = vu. So the units of a ring are the elements in the ring which have multiplicative inverses. Now, we need a way to measure the size of individual elements in a domain. Definition 3 Let R be an integral domain. A function N : R ! Z+ [ 0 with N(0) = 0 is called the norm on the integral domain R. The norm is called a positive norm if N(a) > 0 for all a 6= 0. Also, N(ab) 6 N(a)N(b) for all a, b 2 R, a, b 6= 0. By this definition, any particular integral domain could have several norms. One familiar norm is the absolute value function on the integers. We say that an element has minimum norm if N(u) = ±1. We can now define the units of a ring using the norm. Definition 4 A unit u of a ring R is a nonzero element that has minimum norm, N, N(u) = ±1. In Z, the only units are the integers 1 and -1, since j1j = 1 = j - 1j, where the norm on the integers is the usual absolute value function. Definition 5 An element b in a commutative ring with unity is called an associate of an element a if a = bu for some unit u. Example 10 In Z, 3 and -3 are associates since -3 = 3(-1) and -1 is a unit in Z. So associates differ from each other by a unit. 5 Definition 6 An irreducible element is a nonzero, nonunit element whose only divisors are its associates and the units of the domain. So, an element p of an integral domain is irreducible if and only if whenever p = rs, then r or s is a unit. Example 11 In Z, the irreducible elements are all of the prime integers, since the only divisors of a prime integer are itself and 1 and -1, the units of Z. Definition 7 An integral domain R is a Unique Factorization Domain (UFD) if every nonzero, nonunit element r 2 R has the following two properties: 1.