On Inverses and Linear Independence A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Jeremy S. Moore June 2011 © 2011 Jeremy S. Moore. All Rights Reserved. 2 This dissertation titled On Inverses and Linear Independence by JEREMY S. MOORE has been approved for the Department of Mathematics and the College of Arts and Sciences by Sergio Lopez-Permouth´ Professor of Mathematics Benjamin M. Ogles Dean for College of Arts and Sciences 3 Abstract MOORE, JEREMY S., Ph.D., June 2011, Mathematics On Inverses and Linear Independence (66 pp.) Director of Dissertation: Sergio Lopez-Permouth´ We study various connections between the notions of invertibility of elements and linear independence of subsets of algebras over (not necessarily commutative) rings. The main emphasis is on two such notions: invertible and fluid algebras. We introduce a hierarchy of notions about algebras having a basis B consisting entirely of units. Such a basis is called an invertible basis and algebras that have invertible bases are said to be invertible algebras. The conditions considered in that hierarchy include the requirement that for an invertible basis B, the set of inverses B−1 be itself a basis, the notion that B be closed under inverses and the idea that B be closed under products under a slight commutativity requirement. Among other results, it is shown that this last property is unique of group rings. Many examples are considered and it is determined that the hierarchy is for the most part strict. For any field F , F2, all semisimple F-algebras are invertible. Semisimple invertible F2-algebras are fully characterized. Likewise, the question of which single-variable polynomials over a field yield invertible quotient rings of the F-algebra F[x] is completely answered. Connections between invertible algebras and S-rings (rings generated by units) are also explored. While group rings are the archetype of invertible algebras, this notion is general enough to include many other families of algebras. For example, field extensions and all crossed products (including in particular skew and twisted group rings) are invertible algebras. We consider invertible bases B such that for any two elements from B, a scalar multiple of their product belongs to B. Alternatively, one may consider invertible bases with the requirement that for every basis element, a scalar multiple of its inverse must also be in the basis. We refer to these algebras, respectively, as being scalarly closed under 4 products and scalarly closed under inverses. We explore connections between these ideas and crossed products, twisted group rings, and skew group rings. We show that matrix rings over arbitrary rings are invertible algebras and also determine some types of infinite matrix rings which are invertible too. We conclude the dissertation considering the property that sets of inverses of linearly independent invertible elements be also linearly independent. We refer to algebras with this property as fluid algebras. Fluidity of direct sums will be considered. We will characterize which single-variable polynomials over a field yield fluid quotient algebras of the F-algebra F[x]. We apply those results to establish when finite field extensions are fluid algebras. Also we will show that infinite field extensions are rarely fluid. We then define the fluidity of an R-algebra A to be an integer, such that for every set of n or less linearly independent invertible elements, their inverses are also linearly independent. The fluidity of various families of algebras such as matrix rings and field extensions is explored. Approved: Sergio Lopez-Permouth´ Professor of Mathematics 5 I dedicate this work to all my friends and family. I feel truly blessed to have these people in my life and would not be where or who I am today without them. 6 Acknowledgements I would like to thank Dr. Sergio Lopez-Permouth, my advisor, for his assistance and guidance. I have learned extensively from his knowledge and wisdom of mathematics. Not only that but his dedication, joy, and patience in this process will strengthen me as a mathematician and educator. I would like to thank Kelly, my future wife, who, throughout this process has been nothing short of amazing. Her care and support has been invaluable and words would not be able to express my gratitude. I would like to thank Cindy, Eric and Josh Moore, my mother, father and brother, for their love and understanding during this process and all through my life. I would like to thank Dr. Steve Noltie, who, encouraged me to pursue mathematics as an undergraduate freshman. I would like to thank Dr. Franco Guerriero, for being available and helping with questions as I studied for the algebra comprehensive exam and participating as one of my committee members. I would like to thank the following people for their insight and communication during my studies: Dr. Steve Szabo, Dr. Ryan Botts, Josh Beal, Don Daws, Chris Holston, Ryan Schweibert, Dr. Benigno Parra, Eric Heinzman, Doug Hoffman, and Dr. Ashish Srivastava. In addition, I would like to thank the various faculty members of the Ohio University Mathematics Department. Your assistance and care for students’ success has been admirable. 7 Table of Contents Abstract.........................................3 Dedication........................................5 Acknowledgements...................................6 List of Figures......................................8 1 Preface........................................9 2 Introduction and Preliminaries........................... 11 2.1 Introduction.................................. 11 2.2 Definitions................................... 13 3 Invertible Algebras................................. 16 3.1 Definitions and Preliminary Results..................... 16 3.2 General Results and Some Families of Invertible Algebras......... 21 3.3 Trivial Units.................................. 29 4 Further Results on Invertible Algebras....................... 33 4.1 Infinite Dimensional Invertible Algebras................... 33 4.2 Linear Closed Under Products and Inverses................. 36 4.3 Scalar Closed Under Products and Inverses................. 41 5 Fluidity....................................... 51 5.1 Definitions and Preliminary Results..................... 51 5.2 Fluid Field Extensions and Related Ideas................... 54 5.3 Fluid Matrix Algebras............................. 60 References........................................ 66 8 List of Figures 4.1 Hierarchy of Invertibility............................. 50 9 1 Preface In this dissertation we initiate the study of two new notions regarding the units of an algebra A over a (not necessarily commutative) ring R and the linear independence of its subsets. We call them invertible and fluid algebras. An algebra A over a ring R is said to be an invertible algebra if it has a basis B consisting entirely of units; such a basis is said to be an invertible basis. There are many instances of invertible algebras in the literature. For example, a field extension E of a field F is an example of a case when E is viewed as an algebra over F. In fact, obviously the same is true for division ring extensions. A group ring R[G] over a ring R is a strikingly different example; twisted and skew group rings as well as their common generalization the crossed products are also examples of invertible algebras. When the base ring R is a field, invertible algebras are an example of the S -rings studied in [13] but, in general, the connection does not necessarily hold. In fact, while there is some affinity between the notion of invertible algebras and the various concepts in the literature dealing with rings generated by their units [2], [14], the affinity is only superficial and no real connection seems to exist. Obviously, a notion general enough to encompass so many different types of algebras will be too general to yield very many results since these results would have to be properties that are common to such a diverse collection of examples. For that reason, the reasonable way to proceed is to consider various layers of restrictions on invertible algebras and establish a hierarchy of notions that bring us closer or move us farther away from the known examples. Taking group rings as a motivation, in Chapter3 we consider various such restrictions and find that they constitute a hierarchy. We notice, in particular, that the hierarchy is for the most part strict. Similar conditions but based on the two main examples of crossed products (the skew and twisted group rings) are also considered in Chapter4. A common thread in Chapters3 and4 is the consideration of the extent to 10 which known results about group rings and crossed products extend to invertible algebras. The invertibility of specific families of algebras is also a recurring theme as we discover, for example, that matrix rings and single-variable polynomial factor rings over a field (other than F2) are examples of invertible algebras. A new idea arose during our research of invertible algebras. This idea requires for every set of linearly independent units of an algebra, we have their inverses also remain linearly independent, such algebras we call fluid algebras. Despite the naturality of this notion, there does not seem to be any literature on this type of problem. A motivation for this study was the idea of invertible-2. However, for fluid algebras it is required that for every set of linearly independent invertible elements S, the set of inverses S−1 also be linearly independent, while invertible-2 only asks that there exist one such a set (and that it be a basis!) In fact, a fluid algebra need not even be an invertible algebra. 11 2 Introduction and Preliminaries 2.1 Introduction This dissertation is about the interactions between two topics: invertibility of elements and linear independence of subsets. We have noticed that there does not seem to be much published mathematical research on such connections. First we investigate the idea of algebras having bases consisting entirely of units. We consider various types of these so-called invertible bases.