Exercises in Basic Ring Theory

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1 Exercises in Basic Ring Theory

2 Kluwer Texts in the Mathematical Sciences VOLUME 20 A Graduate-Level Book Series The titles published in this series are listed at the end of this volume.

3 Exercises in Basic Ring Theory by Grigore Calugareanu "Babq-Bolyai" University, Cluj-Napoca, Romania and Peter Hamburg Fernuniversitiit GH, Hagen, Germany Springer-Science+Business Media, B.Y.

4 A C.LP. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOl / Printed on acid-free paper All Rights Reserved 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in Softcover reprint of the hardcover 1 st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 This volume is dedicated to the memory of my father and our professor George Calugareanu to my beloved family Mara, Ilinca, Manole and Daniela and to the rhemory of my mother Zoe CaJugareanu

6 Contents Preface List of Symbols Xl Xlll I EXERCISES 1 1 FUndamentals 3 2 Ideals 9 3 Zero Divisors 15 4 Ring Homomorphisms 19 5 Characteristics 23 6 Divisibility in Integral Domains 27 7 Division Rings 31 8 A utomorphisms 35 9 The Tensor Product Artinian and Noetherian Rings Socle and Radical 45 Vll

7 Vlll CONTENTS 12 Semisimple Rings 13 Prime Ideals, Local Rings 14 Polynomial Rings 15 Rings of Quotients 16 Rings of Continuous Functions 17 Special Problems II SOLUTIONS 1 Fundamentals 2 Ideals 3 Zero Divisors 4 Ring Homomorphisms 5 Characteristics 6 Divisibility in Integral Domains 7 Division Rings 8 Automorphims 9 The Tensor Product 10 Artinian and Noetherian Rings 11 Socle and Radical 12 Semisimple Rings 13 Prime Ideals, Local Rings

8 CONTENTS IX 14 Polynomial Rings Rings of Quotients Rings of Continuous FUnctions Special problems 187 Bibliography 195 Index 197

9 Preface Each undergraduate course of algebra begins with basic notions and results concerning groups, rings, modules and linear algebra. That is, it begins with simple notions and simple results. Our intention was to provide a collection of exercises which cover only the easy part of ring theory, what we have named the "Basics of Ring Theory". This seems to be the part each student or beginner in ring theory (or even algebra) should know - but surely trying to solve as many of these exercises as possible independently. As difficult (or impossible) as this may seem, we have made every effort to avoid modules, lattices and field extensions in this collection and to remain in the ring area as much as possible. A brief look at the bibliography obviously shows that we don't claim much originality (one could name this the folklore of ring theory) for the statements of the exercises we have chosen (but this was a difficult task: indeed, the 28 titles contain approximatively problems and our collection contains only 346). The real value of our book is the part which contains all the solutions of these exercises. We have tried to draw up these solutions as detailed as possible, so that each beginner can progress without skilled help. The book is divided in two parts each consisting of seventeen chapters, the first part containing the exercises and the second part the solutions. For the reader's convenience, each chapter begins with a short introduction giving the basic definitions and results one should know in order to solve the corresponding exercises. Some basic facts concerning groups and modules (vector spaces) are naturally assumed (e.g. cyclic xi

10 xii PREFACE groups, Lagrange theorem etc.). Also simple topology notions and results are assumed especially in the chapter devoted to rings of continuous functions. A small part of this collection with hints and partial solutions, written by the first author, was ready in 1978, but only for internal use for problem sessions. The rest of the 346 exercises were almost all solved in problem sessions in the last 28 years. The chapter with exercises concerning rings of continuous functions is part of a one year course and series of seminars given by the second author. Recently, the authors have become aware of the new publication Exercises in Classical Ring Theory written by T.Y. Lam, an excellent problem book published by Springer-Verlag. With only a few exceptions, this book actually does not contain genuinely simple exercises in ring theory, such as the exercises our collection provides. As Pham Ngoc Anh told us: "this is the book one should have before the problem book written by T.Y. Lam". Needless to say, the intersection of these two collections is nearly void. We therefore strongly hope that, using, in this order, these two books, one should have the best possible start in ring theory. The first author acknowledges his colleague Horia F. Pop, lecturer in Computer Science, for his constant guidance and assistance in using computers and especially Latex and Scientific Word.

11 List of Symbols Symbol IP IN 7l.. Q IR C Mn(R) Eij = E'/j Sn 7l.. n H P(M) Po(M) Rap R[X] R (X) = R[[X]] n(r) Z(R) Id(R) U(R) End(R) ~ut([{) im(j) ker(j) f-l(y) t r Description the set of the prime numbers the set of the non-negative integer numbers the set of the integer numbers the set of the rationals numbers the set of the real numbers the set of the complex numbers the set of the n x n-square matrices with entries in R the matric units the group of the permutations of degree n the ring of the integers modulo n the ring of the quaternions the set of the subsets of M the set of the finite subsets of M the opposite ring of R the ring of the polynomials of indeterminate X over R the ring of the power series of indeterminate X over R the number of the elements of the ring R the center of R the set of the idempotent elements of R the set of the units of R the ring of the endomorphisms of R the group of the automorphisms of K the image of f the kernel of f the preimage of Y by f the left translation with r xiii

12 XIV (X) l(x) r(x) R x R' IIRi iei 51 EB S2 char(r) M RN s(r) rad(r) R(R) N(R) Spec(R) Rs Rp Q(R) TO TO C(X) C*(X) Z(f) Z(X) XA Sats(I) RX A:B v7 Q(p) V(X) the ideal generated by X the left annihilator of X the right annihilator of X the direct product the direct product the direct sum the characteristics of R LIST OF SYMBOLS the tensor product of R-modules the socle of R the (Jacobson) radical of R the prime radical of R the nilradical of R the prime spectrum the ring of quotients the localization of R in P the classical (total) ring of quotients the discrete topology the indiscrete topology the continuous real-valued functions on X the bounded real-valued functions on X the constant function with value a the zero-set associated to I the set of the zero-sets of the space X a characteristic function = {r E RI3s E S : rs E I} = {III: X -+ R} = {r E RIVb E B : rb E A} = {r E RI3n E IN : rn E I} ={~I(n;p)=l} = {P E 5pec(R)IX ~ P}

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