Problem Books in Mathematics
Edited by Peter Winkler
Problem Books in Mathematics Series Editors: Peter Winkler
Pell's Equation by Edward J. Barbeau Polynomials by Edward J. Barbeau Problems in Geometry by Marcel Berger, Pierre Pansu, Jean-Pic Berry, and Xavier Saint-Raymond Problem Book for First Year Calculus by George W. Bluman Exercises in Probability by T. Cacoullos Probability Through Problems by Marek Capinski and Tomasz Zastawniak
An Introduction to Hilbert Space and Quantum Logic by David W. Cohen Unsolved Problems in Geometry by Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy
Berkeley Problems in Mathematics (Third Edition) by Paulo Ney de Souza and Jorge-Nuno Silva The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004 by Dusan Djukic, Vladimir Z. Jankovic, Ivan Matic, and Nikola Petrovic Problem-Solving Strategies by Arthur Engel Problems in Analysis by Bernard R. Gelbaum Problems in Real and Complex Analysis by Bernard R. Gelbaum (continued after subject index)
T.Y. Lam
Exercises in Modules and Rings
~ Springer
T.Y. Lam Department of Mathematics University of California, Berkeley Berkeley, CA 94720-0001 USA
[email protected]
Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA Peter.
[email protected]
Mathematics Subject Classification (2000): 00A07 13-XX 16-01 Library of Congress Control Number: 2006935420 ISBN-lO: 0-387-98850-5 ISBN-13: 978-0-387-98850-4
e-ISBN-10: 0-387-48899-5 e-ISBN-13: 978-0-387-48899-8
Printed on acid-free paper.
©
2007 Springer Science+ Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
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To Chee King A constant source of strength and inspiration
Preface
The idea of writing this book came roughly at the time of publication of my graduate text Lectures on Modules and Rings, Springer GTM Vol. 189, 1999. Since that time, teaching obligations and intermittent intervention of other projects caused prolonged delays in the work on this volume. Only a lucky break in my schedule in 2006 enabled me to put the finishing touches on the completion of this long overdue book. This book is intended to serve a dual purpose. First, it is designed as a "problem book" for Lectures. As such, it contains the statements and full solutions of the many exercises that appeared in Lectures. Second, this book is also offered as a reference and repository