Exercises in Classical Ring Theory

380 Pages·2003·1.521 MB·English

by  T.Y. Lam

Problem Books in Mathematics
Edited by K.A. Bencs´ath
P.R. Halmos
Springer
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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
[e-mail]
Series Editors:
Katalin A. Bencs´ath Paul R. Halmos
Mathematics Department of Mathematics
School of Science Santa Clara University
Manhattan College Santa Clara, CA 95053
Riverdale, NY 10471 USA
USA [e-mail]
[e-mail]
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
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To Chee King
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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
[e-mail].
Berkeley, California T.Y.L.
01/02/03
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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