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Problem Books in Mathematics Edited by K.A. Bencsáth P.R. Halmos Springer NewYork Berlin Heidelberg HongKong London Milan Paris Tokyo This page intentionally left blank T.Y. Lam Exercises in Classical Ring Theory Second Edition T.Y. Lam Department of Mathematics University of California, Berkeley Berkeley, CA 94720-0001 USA [email protected] Series Editors: Katalin A. Bencsáth Paul R. Halmos Mathematics Department of Mathematics School of Science Santa Clara University Manhattan College Santa Clara, CA 95053 Riverdale, NY 10471 USA USA [email protected] [email protected] MathematicsSubjectClassification(2000):00A07,13-01,16-01 LibraryofCongressCataloging-in-PublicationData Lam,T.Y.(Tsit-Yuen),1942– Exercisesinclassicalringtheory/T.Y.Lam.—2nded. p.cm.—(Problembooksinmathematics) Includesindexes. ISBN0-387-00500-5(alk.paper) 1.Rings(Algebra) I.Title.II.Series. QA247.L262003 512(cid:1).4—dc21 2003042429 ISBN0-387-00500-5 Printedonacid-freepaper. @2003,1994Springer-VerlagNewYork,Inc. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout thewrittenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAv- enue,NewYork,NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now knownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms, eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionas towhetherornottheyaresubjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 SPIN10913086 www.springer-ny.com Springer-Verlag NewYork Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH To Chee King Juwen, Fumei, Juleen, and Dee-Dee This page intentionally left blank Preface to the Second Edition The four hundred problems in the first edition of this book were largely based on the original collection of exercises in my Springer Graduate Text AFirstCourseinNoncommutativeRings,ca.1991.Asecondeditionofthis ring theory text has since come out in 2001. Among the special features of thiseditionwastheinclusionofalargenumberofnewlydesignedexercises, many of which have not appeared in other books before. It has been my intention to make the solutions to these new exercises available. Since Exercises in Classical Ring Theory has also gone out of print recently, this seemed a propitious time to issue a new edition of our Problem Book. In this second edition, typographical errors were corrected, variousimprovementsonproblemsolutionsweremade,andtheComments on many individual problems have been expanded and updated. All in all, we have added eighty-five exercises to the first edition, some of which are quite challenging. In particular, all exercises in the second edition of First Course are solved here, with essentially the same reference numbers. As before, we envisage this book to be useful in at least three ways: (1) as a companion to First Course (second edition), (2) as a source book for self-study in problem-solving, and (3) as a convenient reference for much ofthefolkloreinclassicalringtheorythatisnoteasilyavailableelsewhere. Hearty thanks are due to my U.C. colleagues K. Goodearl and H.W. Lenstra, Jr. who suggested several delightful exercises for this new edition. While we have tried our best to ensure the accuracy of the text, occa- sional slips are perhaps inevitable. I’ll welcome comments from my read- ers on the problems covered in this book, and invite them to send me their corrections and suggestions for further improvements at the address [email protected]. Berkeley, California T.Y.L. 01/02/03 This page intentionally left blank Preface to the First Edition This is a book I wished I had found when, many years ago, I first learned the subject of ring theory. All those years have flown by, but I still did not find that book. So finally I decided to write it myself. All the books I have written so far were developed from my lectures; this one is no exception. After writing A First Course in Noncommutative Rings(Springer-VerlagGTM131,hereafterreferredtoas“FC”),Itaught ringtheoryinBerkeleyagaininthefallof1993,usingFCastext.Sincethe main theory is already fully developed in FC, I asked my students to read thebookathome,sothatwecouldusepartoftheclasstimefordoingthe exercisesfromFC.Thecombinationoflecturesandproblemsessionsturned outtobeagreatsuccess.Bytheendofthecourse,wecoveredasignificant portion of FC and solved a good number of problems. There were 329 exercises in FC; while teaching the course, I compiled 71 additional ones. Theresultingfourhundredexercises,withtheirfullsolutions,comprisethis ring theory problem book. There are many good reasons for a problem book to be written in ring theory,orforthatmatterinanysubjectofmathematics.First,thesolutions to different exercises serve to illustrate the problem-solving process and showhowgeneraltheoremsinringtheoryareappliedinspecialsituations. Second, the compilation of solutions to interesting and unusual exercises extendsandcompletesthestandardtreatmentofthesubjectintextbooks. Last, but not least, a problem book provides a natural place in which to recordleisurelysomeofthefolkloreofthesubject:the“tricksofthetrade” inringtheory,whicharewellknowntotheexpertsinthefieldbutmaynot befamiliartoothers,andforwhichthereisusuallynogoodreference.With all of the above objectives in mind, I offer this modest problem book for