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UC San Diego UC San Diego Electronic Theses and Dissertations UC San Diego UC San Diego Electronic Theses and Dissertations Title Fun with tensor products Permalink https://escholarship.org/uc/item/91z3d9c8 Author Horn, Larissa Dawn Publication Date 2008 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, SAN DIEGO Fun with Tensor Products A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Larissa Dawn Horn Committee in charge: Professor Lance Small, Chair Professor Dan Rogalski Professor Paul Siegel Professor Jack Wolf Professor Efim Zelmanov 2008 Copyright Larissa Dawn Horn, 2008 All rights reserved. The dissertation of Larissa Dawn Horn is approved, and it is acceptable in quality and form for publication on microfilm: Chair University of California, San Diego 2008 iii DEDICATION To all the friends, family, and loved ones who been here with me. iv EPIGRAPH In the sweetness of friendship let there be laughter, and sharing of pleasures. For in the dew of little things the heart finds its morning and is refreshed. —Kahlil Gibran v TABLE OF CONTENTS Signature Page . iii Dedication . iv Epigraph . v Table of Contents . vi Acknowledgements . vii Vita and Publications . viii Abstract . ix 1 Introduction . 1 2 Division Algebras . 5 2.1 PI Algebras . 5 2.2 Kaplansky’s Theorem . 6 2.3 A Second Proof . 7 2.4 The Algebraic Case . 8 2.5 The Transcendental Case . 9 3 The Martindale Ring of Quotients . 11 3.1 Definitions . 12 3.2 Examples of Centrally Closed Algebras . 13 3.3 Properties of Centrally Closed Algebras . 14 3.4 The Polynomial Ring Over a Centrally Closed Algebra . 15 4 Jacobson Rings . 16 4.1 Tensor Products with Simple Rings . 16 4.2 Tensoring with Centrally Closed Primitive Rings . 18 5 Krull Dimension . 19 5.1 Krull Dimension 0 . 20 5.2 Krull Dimension 1 . 21 5.3 Higher Krull Dimension and Other Questions . 23 Bibliography . 25 vi ACKNOWLEDGEMENTS I would like to first thank my advisor, Lance Small, for both mental and moral support over the past years. I would also like to thank Dan Rogalski for his many illustrative comments and John Farina for always being there to talk math and help translate Lance’s comments. And finally, I would like to thank all the students and faculty in the math department that make this such a welcoming place to learn alongside others. Not least amongst those are my dear friends. I thank Kristin for enjoying the sanity of the outdoors with me. I thank Amanda for the amazing times we have had when allowing a bit of insanity. And to the rest of the crowd, I am ever grateful for the conversations, the hikes, the bike rides, and the fellowship you have all given me. I thank my brother, Aaron, who helped me realize that there are many ways to be successful in life. And I thank Jon, who taught me that living a thoughtful life is one of them. I thank my parents, who gave me the independence and strong will to accomplish the things I desire. Finally, I thank Mike, who has made this year more wonderful than I ever expected. Your love means the world to me. vii VITA 2003 B. S. in Mathematics magna cum laude, Emory University 2003-2008 Graduate Teaching Assistant, University of California, San Diego 2008 Ph. D. in Mathematics, University of California, San Diego viii ABSTRACT OF THE DISSERTATION Fun with Tensor Products by Larissa Dawn Horn Doctor of Philosophy in Mathematics University of California San Diego, 2008 Professor Lance Small, Chair This dissertation addresses various questions in noncommutative ring theory that may be solved with tensor product constructions. First, we consider division rings finite dimensional over their center. Using Ka- plansky’s Theorem one may show that any subdivision ring will also be finite dimensional over its center. We present an unpublished proof of this theorem, which generalizes to a new result: If D is a division ring algebraic over its center, Z, then any subdivision ring, D0, will also be algebraic over its center, Z0. In the following chapter we look at the Martindale ring of quotients of a ring. A ring, A, is called centrally closed if the center of its Martindale ring of quotients is equal to the center of A. If A is centrally closed, then any ideal of A ⊗k B will contain an ideal of the form I ⊗k J. We extend this result to show that prime ideals of A ⊗k B contain ideals of the form P ⊗k B or A ⊗k Q for P and Q prime ideals of A and B, respectively. Next, we consider Jacobson rings, and we show that a simple ring tensored with a Jacobson ring remains Jacobson. This determines new classes of Jacobson rings, broadening results of Jordan, Goodearl and Warfield. Finally, we look at the classical Krull dimension of polynomial rings over a noncommutative coefficient ring. We show that if R is centrally closed and has classical Krull dimension 1, then R[x] must have classical Krull dimension 2. ix 1 Introduction The idea of a tensor product arises naturally in algebra as a universal object in the category of multilinear maps of a fixed set of modules. As we will see, there are many interesting applications of this construction. First, we look at some basic definitions and properties. In this paper, all rings will contain an identity element. Let R be a ring and VR and RW right and left R-modules, respectively. For A, an abelian group, a bilinear map ϕ : V × W → A is said to be balanced if we have ϕ(va × w) = ϕ(v × aw) for all a ∈ R, v ∈ V , and w ∈ W . We define the tensor product of V and W over R to be an abelian group V ⊗R W along with the balanced map i : V × W → V ⊗R W that satisfies the following universal criterion: For any balanced map from V × W to an abelian group, ϕ : V × W → A, there exists a unique homomorphism χ : V ⊗k W → A such that ϕ = χ ◦ i. That is, the following diagram commutes. i V × W / V ⊗R W MMM MMM χ ϕ MM MMM & A More concretely, we may define V ⊗RW as the quotient of a free group by certain relations. Set F = hV ×W i, the free abelian group on the set {(v, w)|v ∈ V and w ∈ W }. We will subsequently denote the element (v, w) ∈ F as v ⊗ w. Let H be the subgroup of F generated by the following elements: • (v1 + v2) ⊗ w − v1 ⊗ w − v2 ⊗ w for all v1, v2 ∈ V and w ∈ W , • v ⊗ (w1 + w2) − v ⊗ w1 − v ⊗ w2 for all v ∈ V and w1, w2 ∈ W , and • v ⊗ rw − vr ⊗ w for all v ∈ V , w ∈ W , and r ∈ R. 1 2 Then, we may define V ⊗R W to be the quotient F/H. If furthermore, our modules are algebras over a commutative ring or field, k, we may describe the tensor product as a k-algebra in the following way. For A and B k-algebras, we define multiplication in the tensor product by a1 ⊗b1 ·a2 ⊗b2 = a1a2 ⊗b1b2. Since both A and B are k-algebras, α · a ⊗k b = αa ⊗k b = a ⊗k αb. In particular, for a simple ring, S, with center equal to the field k, we may tensor S over k with any other k-algebra. Note that if the field we are tensoring over is clear, we may drop the subscript and simply write A ⊗ B. Before we come to examples and properties, we should make a few comments on the tensor product construction. On the one hand, taking the free group on all ordered pairs from A and B gives a very large object; however, modding out by the relations may completely collapse the algebra. Understanding the resulting structure is often tricky and a very important part of algebra. There are but a few cases where we completely understand the structure of a tensor product. Some of the better understood examples of tensor products involve simple rings, polynomial rings, or matrix rings. For example, we can extend the coefficients of a polynomial ring by tensoring up to a larger ring. That is, if R embeds in S, then ∼ ∼ R[x] ⊗R S = S[x]. Similarly, for matrices over a k-algebra, A, we can write Mn(A) = Mn(k) ⊗k A. In that tensor products often yield interesting objects of study, a natural ques- tion arises: How are structure-theoretic properties determined by the underlying rings? We would like to know which properties will pass through the construction. To this end, one of the most important problems is to find constraints on the ideal structure of A ⊗R B. Although I ⊗R J is an ideal of A ⊗R B whenever I and J are ideals of A and B, it is not generally true that all ideals are of this form. There are positive results only in certain cases. For central simple algebras the correspondence is perfect. An algebra, A, is central simple over a field k if A is simple and the center of A, Z(A), is exactly k. If A is a central simple algebra over k and B is any other k-algebra, then the ideals of A ⊗k B are always of the form A ⊗k I for I an ideal of B. Similarly, prime ideals are of the form A ⊗k P for P a prime ideal of B.
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