Yurij A. Drozd Vladimir V. Kirichenko Finite Dimensional Algebras With an Appendix by Vlastimil Dlab Translated from the Russian by Vlastimil Dlab Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Yurij A. Drozd Vladimir V. Kirichenko Faculty of Mechanics and Mathematics Kiev Taras Shevchenko University 252617 Kiev, Ukraina Translator: Vlastimil Dlab Department of Mathematics and Statistics Carleton University Ottawa K1S 5B6, Canada Title of the Russian edition: Konechnomemye algebry. Publisher of Kiev State University, Vishcha Shkola, Kiev 1980 Mathematics Subject Classification (1991): 16P10, 16GlO, 16K20 ISBN-13:978-3-642-76246-8 e-ISBN-13:978-3-642-76244-4 DOl: 10.1007/978-3-642-76244-4 Cataloging-in-Publication Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965. in its current version. and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Softcover reprint of the hardcover 1st edition 1994 Typesetting: Camera-ready copy from the translator SPIN 10028296 4113140 - 5 432 I 0 - Printed on acid-free paper Preface to the English Edition This English edition has an additional chapter "Elements of Homological Al­ gebra". Homological methods appear to be effective in many problems in the theory of algebras; we hope their inclusion makes this book more complete and self-contained as a textbook. We have also taken this occasion to correct several inaccuracies and errors in the original Russian edition. We should like to express our gratitude to V. Dlab who has not only metic­ ulously translated the text, but has also contributed by writing an Appendix devoted to a new important class of algebras, viz. quasi-hereditary algebras. Finally, we are indebted to the publishers, Springer-Verlag, for enabling this book to reach such a wide audience in the world of mathematical community. Kiev, February 1993 Yu.A. Drozd V.V. Kirichenko Preface The theory of finite dimensional algebras is one of the oldest branches of modern algebra. Its origin is linked to the work of Hamilton who discovered the famous algebra of quaternions, and Cayley who developed matrix theory. Later finite dimensional algebras were studied by a large number of mathematicians including B. Peirce, C.S. Peirce, Clifford, ·Weierstrass, Dedekind, Jordan and Frobenius. At the end of the last century T. Molien and E. Cartan described the semisimple algebras over the complex and real fields and paved the first steps towards the study of non-semi simple algebras. A new period in the development of the theory of finite dimensional al­ gebras opened with the work of Wedderburn; the fundamental results of this theory belong to him: a description of the structure of semis imp Ie algebras over an arbitrary field; the theorem on lifting the quotient algebra by its rad­ ical; the theorem on commutativity of finite division rings, etc. Most of his results were extended to rings with minimal condition (artinian rings) and the theory of semi simple algebras found its present form in the work of algebraists from the German school headed by E. Noether, E. Artin and R. Brauer. More­ over, their work exposed the fundamental role of the concept of a module (or representation) . Further development of the theory advanced basically in two directions. The first led to establishing the theory of infinite dimensional algebras (and rings without chain conditions); those results are reflected in the monograph "Structure of rings" by N. Jacobson. The second direction - the study of the structure of non-semisimple algebras - met considerable difficulties, most of which were not overcome till now. Therefore papers which single out and describe "natural" classes of algebras occupy here an important place. This direction originates in the investigations of Kothe, Asano and Nakayama of principal ideal algebras and their generalizations. Throughout its development, the theory of finite dimensional algebras was closely related to various branches of mathematics, acquiring from them new ideas and methods, and in turn exerting influence on their development. In the initial period, the most profound connections were to linear algebra, the theory of groups and their representations and Galois theory. Recently, in particular in connection with the study of non-semisimple algebras, an important role is being played by methods of homological algebra, category theory and algebraic geometry. viii Preface The present book is intended to be a modern textbook on the theory of finite dimensional algebras. The basic tools of investigation are methods of the theory of modules (representations), which, in our opinion, allow a very simple and clear approach both to classical and new results. Naturally, we cannot pursue all directions equally. The principal goal of this book is structure theory, i. e. investigation of the structure of algebras. In particular, the general theory of representations of algebras, which is presently undergoing a remarkable resurgence, is almost not touched upon. Undoubtedly, specialists will notice the absence of some other areas recently developed. Nevertheless, we hope that the present book will enable the reader to learn the basic results of the classical theory of algebras and to acquire sufficient background to follow and be able to get familiar with contemporary inv~stigations. A large portion of the book is based on the standard university course in abstract and linear algebra and is fully accessible to students of the second and third year. In particular, we do not assume knowledge of any preliminary information on the theory of rings and modules (moreover, the word "ring" is almost absent in the book). The chapters devoted to group representations and Galois theory require, of course, familiarity with elements of group theory (for instance, to the extent of A.1. Kostrikin's textbook "An introduction to algebra"). At the end of each chapter, we provide exercises of varied complexity which contain examples instrumental for understanding the material as well as fragments of theories which are not reflected in the main text. We strongly recommend that readers (and in particular, beginners) work through most of the exercises on their first reading. The most difficult ones are accompanied by rather explicit hints. The content of the book is divided into three parts. The first part consists of Chapters 1-3; here the basic concepts of the theory of algebras are discussed, and the classical theory of semisimple algebras and radicals is explained. The second part, Chapters 4-6, can be called the "subtle theory of semisimple algebras". Here, using the technique of tensor products and bimodules, the theory of central simple algebras, elements of Galois field theory, the concept of the Brauer group and the theory of separable algebras are presented. Finally, the third part, Chapters 8-10, is devoted to more recent results: to the Morita theorem on equivalence of module categories, to the theory of quasi-Frobenius, uniserial, hereditary and serial algebras. Some of the results of these latter chapters until now have been available only in journal articles. A somewhat special place is occupied by Chapter 7; in it are developed, based on the results from semisimple algebras, the theory of group representations up to the integral theorems and the Burnside theorem on solvability of a group of order paqb. Naturally, we have not tried to formulate and prove theorems in their ut­ most generality. Besides, we have used the fact that we deal only with the finite dimensional case whenever, in our view, it simplified our presentation. An experienced reader will certainly note that many results hold, for example, for arbitrary artinian rings. Very often such generalizations follow almost au­ tomatically, and when this is not the case, it would be necessary to introduce Preface ix sufficiently complex new concepts, which, in our opinion, could make reading the book substantially more difficult. We are not presenting a complete list of references on finite dimensional algebras because, even when restricted to topics covered in the book, it would probably be comparable in length to the entire work. We point out only sev­ eral textbooks and monographs in which the reader can get acquainted with other aspects of the theory of rings and algebras [1,2,4-9]. The questions of arithmetic of semisimple algebras are dealt with in the book [10], or in the classical textbook of Deuring [3]. We follow generally used notation. In particular, the symbols <Q, JR, {: de­ note, respectively, the fields of rational, real and complex numbers. Numbering of statements is done in the book by sections. For instance, "Theorem 4.6.5" denotes the fifth theorem in Section 6 of Chapter 4. Contents 1. Introduction ........................................ 1 1.1 Basic Concepts. Examples . 1 1.2 Isomorphisms and Homomorphisms. Division Algebras .. 6 1.3 Representations and Modules. 9 1.4 Submodules and Factor Modules. Ideals and Quotient Algebras . 13 1.5 The Jordan-Holder Theorem .......................... 18 1.6 Direct Sums. 21 1.7 Endomorphisms. The Peirce Decomposition. .. 23 Exerises to Chapter 1 .. .. 29 2. Semisimple Algebras. 31 2.1 Schur's Lemma. 31 2.2 Semisimple Modules and Algebras ...................... 32 2.3 Vector Spaces and Matrices . .. 35 2.4 The Wedderburn-Artin Theorem ....................... 38 2.5 Uniqueness of the Decomposition . 39 2.6 Representations of Semisimple Algebras .................. 40 Exercises to Chapter 2 .............................. 43 3. The Radical. 44 3.1 The Radical of a Module and of an Algebra.