CALIFORNIA STATE UNIVERSITY, NORTHRIDGE NONCOMMUTATIVE RINGS: A SURVEY OF DIVISION RINGS AND SIMPLE RINGS A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Mathematics by Sharon Paesachov December 2012 The thesis of Sharon Paesachov is approved: |||||||||||||||||| |||||||| Dr. Katherine Stevenson Date |||||||||||||||||| |||||||| Dr. Mary Rosen Date |||||||||||||||||| |||||||| Dr. Jerry Rosen, Chair Date California State University, Northridge ii Dedications To my Mom, Kazia and Turunj, and Eran Thanks for all your support iii ACKNOWLEDGEMENTS I would like to take this time to thank all those who have made this work possible. First, to the Chair of my thesis committee, Jerry Rosen, for providing support and guidance. I appreciate all the time spent explaining, and sometimes re-explaining. Second, Mary Rosen, for all your proof reading and editing. Many thanks Katherine Stevenson for serving on my committee. I also want to thank all the students who have struggled with me. You are all like family to me. Without you I just don't know how I would have made it. Thanks Mike Hubbard, Alex Sherbetjian, Shant Mahserejian, Greg Cossel, Spencer Gerhardt, and to a couple who have moved on, Jessie Benavides and Chad Griffith. Finally to my family. I want to thank my mother for all her love and support. I would have never gone back to school without your encouragement. To Kazia, thank you for putting up with all the late, late nights and all the ups and downs, I'm sure it wasn't easy. iv Table of Contents Signature Page . ii Dedications . iii Acknowledgments . iv Abstract . vii Introduction . .1 Chapter 1 Classical Results in Noncommutative Ring Theory . .3 1.1 Modules . .3 1.2 Jacobson Radical . .7 1.3 Artinian Rings . 13 1.4 Semisimple Artinian rings . 20 1.5 Wedderburn-Artin . 25 Chapter 2 Division Rings . 31 2.1 Power series and Laurent series . 31 2.2 Ring of Fractions and the Ore Domain . 37 2.3 Noetherian Rings . 42 2.4 Skew Polynomial Rings . 46 2.5 Generic Ring of Matrices . 52 2.6 Goldie Rings . 55 Chapter 3 Simple Rings . 62 2.1 Construction of Simple Rings . 62 2.2 Central Localization . 68 2.3 Examples of Simple Rings . 71 v Bibliography . 81 vi Abstract NONCOMMUTATIVE RINGS: A SURVEY OF DIVISION RINGS AND SIMPLE RINGS by Sharon Paesachov Master of Science in Mathematics In this thesis we start with some important classical results in noncommutative ring theory. Namely, we classify all semisimple Artinian rings in terms of matrices with entries from division rings. In the second chapter we start with some natural constructions of division rings. We do this by taking a polynomial ring and "skewing" the multiplication. Further in the chapter we show what conditions a ring must meet in order to have (or be imbedded in) a division ring of fractions. The remainder of the second chapter is devoted to more constructions of division rings. In the final chapter we move our focus to Simple rings. We give a construction of simple rings as well as many examples. vii INTRODUCTION The primary goal of this thesis is to give methods for constructing two classes of noncommutative rings called division rings and simple rings. A division ring is a ring in which every non-zero element has a multiplicative inverse. The key thing is to note that multiplication need not be commutative. The first example of a (noncommutative) division ring was discovered in 1843 by the English mathematician Sir William Hamilton. Hamilton was searching for a way to represent vecotrs in space in an analogous manner to how vectors in the plane are represented by complex numbers. It turns out such a construction is impossible, but in his failed attempt, Hamilton discovered the Quarternions. Hamilton's discovery turned out to be a division ring as well as a four dimensional vector space over the real numbers. The next example of a division ring was found in 1903 by Hilbert. Hilbert started with the field of Laurent series F ((x)), over the field F = R(t) (the rational function field over the real numbers. He then \skewed" the multiplication via the automorphism which maps t to 2t. That is, the indeterminate x no longer commutes with the coefficients. Instead, we define xt = 2tx. The resulting ring is denoted F ((s; σ)), where σ(t) = 2t. Hilbert showed that F ((s; σ)) is a division ring, called the division ring of skew Laurent series. One interesting facet of Hilbert's division ring is that it is infinite dimensional over its center. In the 1920s and 30s ring theorists developed structure theories for large classes of noncommutative rings. It was discovered that, in some sense, division rings provide the underpinnings for many important classes of noncommutative rings and various division ring constructions were discovered. Some of these constructions are not 1 easily accessible to most people studying advanced math. However, skew Laurent construction only requires a field and an automorphism. We will review the skew Laurent series construction and we will determine the center in all cases (i.e. where the automorphism has finite and infinite periods). We then show how these constructions can be generalized to the case where the coefficient ring is simple (a ring R is simple if its only two-sided ideals are (0) and R). That is, if R is simple and σ is an automorphism of R, then we prove that the skew Laurent series ring R((x; σ)) is simple and we determine its center in all cases. We also show how certain classes on noncommutative domains (Ore domains) have division rings of fractions and use this to give alternative methods for constructing division rings. Finally, we construct simple rings by \skewing" the multiplication via a derivation, instead of an automorphism. We begin the thesis by reviewing some basic noncommutative ring theory so that the reader will get an idea how division rings and simple rings figure into the basic structure theory of noncommutative rings. 2 Chapter 1: Classical Results in Noncommutative Ring Theory We begin our discussion with some classical results in Noncommutative Ring Theory. We start with modules and these lead to our first example of a division ring. Next, our discussion takes us into the exploration of the Jacobson radical of a ring. We will try to find rings that have a \nice" Jacobson radical and, this will allow us to define a semisimple ring and from there to go further and define a simple ring. All of this serves two purposes: to prove the classical theorems of Wedderburn and Artin which state that every simple Artinian ring is isomorphic to a matrix ring over a division ring and that every semisimple Artinian ring is isomorphic to a finite direct product of matrix rings over division rings. These theorems indicate the signficant role played by division rings. Section 1: Modules Definition 1.a: An R-module is a vector space over a ring R. Definition 1.b: Alternatively, we say that the additive abelian group M is said to be an R-module if there exists a mapping M × R −! M defined by (m; r) 7! mr (m 2 M; r 2 R) such that: (1) m(a + b) = ma + mb (2) (m + n)a = ma + na (3) (ma)b = m(ab) for all m; n 2 M and all a; b 2 R. We remark that we are omitting the module axiom which states that m1 = m for all 3 m 2 M where 1 is the unity of R. In this chapter, we are not assuming that our rings contain a unity. In fact, the goal of several of our results is to conlude certain classes of rings do caontain a unity. Since it will be necessary to check that certain objects are submodules, we give the Submodule Criterion as the definition for what a submodule is. Definition 2: Let R be a ring and M an R-module. A subset N of M is a submodule of M if (1) N 6= ; (2) x + yr 2 N for all r 2 R and all x; y 2 N. Definition 3: An R-module M is said to be faithful if Mr = (0) implies r = 0 Definition 4: If M is an R-module, then the set A(M) = fx 2 R : Mx = (0)g. With respect to the definition of faithful, we can now say that M is faithful if A(M) = (0) Lemma 5: A(M) is a two-sided ideal of R. Moreover, M is a faithful R=A(M)-module. Proof: A(M) is clearly an additive subgroup of R. Let r 2 R and x 2 A(M). Now Mxr = (0)r = (0) ) xr 2 A(M). Thus A(M) is a right ideal. To see that A(M) is a left ideal, notice that M(rx) = (Mr)x ⊂ Mx = (0) ) rx 2 A(M). Hence, A(M) is a two-sided ideal of R. To see that M is an R=A(M)-module, let m 2 M; r + A(M) 2 R=A(M). We define the action m(r + A(M)) = mr. To show this is well-defined: 4 r + A(M) = r0 + A(M) ) r − r0 2 A(M) ) m(r − r0) = 0 for all m 2 M ) mr = mr0: Thus 0 0 m(r + A(M)) = mr = mr = m(r + A(M)). Now let m 2 M; r1; r2 2 R, then m(r1 + A(M) + r2 + A(M)) = m(r1 + r2 + A(M)) = m(r1 + r2) = mr1 + mr2 = m(r1 + A(M)) + m(r2 + A(M)): The second and third axioms for modules follow just as easily.