Table Of ContentAlgebras, Rings and Modules
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume575
Algebras, Rings and
Modules
Volume 1
by
Michiel Hazewinkel
CWI,
Amsterdam, The Netherlands
Nadiya Gubareni
Technical University of Częstochowa,
Poland
and
V.V. Kirichenko
Kiev Taras Shevchenko University,
Kiev, Ukraine
KLUWER ACADEMIC PUBLISHERS
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Table of Contents
Preface.....................................................................ix
Chapter 1. Preliminaries..................................................1
1.1 Basic concepts and examples..............................................1
1.2 Modules and homomorphisms............................................15
1.3 Classical isomorphism theorems..........................................18
1.4 Direct sums and products................................................21
1.5 Finitely generated and free modules......................................24
1.6 Notes and references.....................................................27
Chapter 2. Decompositions of rings....................................30
2.1 Two-sided Peirce decompositions of a ring................................30
2.2 The Wedderburn-Artin theorem..........................................33
2.3 Lattices. Boolean algebras and rings.....................................37
2.4 Finitely decomposable rings..............................................50
2.5 Notes and references.....................................................57
Chapter 3. Artinian and Noetherian rings.............................59
3.1 Artinian and Noetherian modules and rings..............................59
3.2 The Jordan-Ho¨lder theorem..............................................64
3.3 The Hilbert basis theorem...............................................67
3.4 The radical of a module and a ring.......................................68
3.5 The radical of Artinian rings.............................................71
3.6 A criterion for a ring to be Artinian or Noetherian.......................74
3.7 Semiprimary rings.......................................................76
3.8 Notes and references.....................................................77
Chapter 4. Categories and functors.....................................82
4.1 Categories, diagrams and functors........................................82
4.2 Exact sequences. Direct sums and direct products........................85
4.3 The Hom functors.......................................................90
4.4 Bimodules...............................................................93
4.5 Tensor products of modules..............................................94
4.6 Tensor product functor...................................................99
4.7 Direct and inverse limits................................................102
4.8 Notes and references....................................................109
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vi ALGEBRAS, RINGS AND MODULES
Chapter 5. Projectives, injectives and flats...........................111
5.1 Projective modules.....................................................111
5.2 Injective modules.......................................................115
5.3 Essential extensions and injective hulls..................................125
5.4 Flat modules...........................................................131
5.5 Right hereditary and right semihereditary rings.........................135
5.6 Herstein-Small rings....................................................139
5.7 Notes and references....................................................141
Chapter 6. Homological dimensions...................................143
6.1 Complexes and homology. Free resolutions..............................143
6.2 Projective and injective resolutions. Derived functors...................146
6.3 The functor Tor........................................................150
6.4 The functor Ext........................................................153
6.5 Projective and injective dimensions.....................................155
6.6 Global dimensions......................................................158
6.7 Notes and references....................................................159
Chapter 7. Integral domains...........................................161
7.1 Principal ideal domains.................................................161
7.2 Factorial rings..........................................................164
7.3 Euclidean domains......................................................169
7.4 Rings of fractions and quotient fields....................................171
7.5 Polynomial rings over factorial rings....................................174
7.6 The Chinese remainder theorem........................................177
7.7 Smith normal form over a PID..........................................178
7.8 Finitely generated modules over a PID..................................181
7.9 The Frobenius theorem.................................................185
7.10 Notes and references...................................................187
Chapter 8. Dedekind domains..........................................189
8.1 Integral closure.........................................................189
8.2 Dedekind domains......................................................193
8.3 Hereditary domains.....................................................199
8.4 Discrete valuation rings.................................................201
8.5 Finitely generated modules over Dedekind domains.....................205
8.6 Pru¨fer rings.............................................................208
8.7 Notes and references....................................................209
Chapter 9. Goldie rings.................................................210
9.1 Ore condition. Classical rings of fractions...............................210
9.2 Prime and semiprime rings..............................................214
9.3 Goldie rings. The Goldie theorem.......................................219
9.4 Notes and references....................................................224
TABLE OF CONTENTS vii
Chapter 10. Semiperfect rings.........................................226
10.1 Local and semilocal rings..............................................226
10.2 Noncommutative discrete valuation rings...............................229
10.3 Lifting idempotents. Semiperfect rings.................................233
10.4 Projective covers. The Krull-Schmidt theorem.........................237
10.5 Perfect rings...........................................................243
10.6 Equivalent categories..................................................248
10.7 The Morita theorem...................................................255
10.8 Notes and references...................................................260
Chapter 11. Quivers of rings...........................................262
11.1 Quivers of semiperfect rings............................................262
11.2 The prime radical......................................................269
11.3 Quivers (finite directed graphs)........................................272
11.4 The prime quiver of a semiperfect ring.................................281
11.5 The Pierce quiver of a semiperfect ring................................ 285
11.6 Decompositions of semiperfect rings....................................288
11.7 The prime quiver of an FDD-ring......................................291
11.8 The quiver associated with an ideal....................................293
11.9 The link graph of a semiperfect ring...................................296
11.10 Notes and references..................................................298
Chapter 12. Serial rings and modules.................................300
12.1 Quivers of serial rings..................................................300
12.2 Semiperfect principal ideal rings.......................................302
12.3 Serial two-sided Noetherian rings......................................304
12.4 Properties of serial two-sided Noetherian rings.........................313
12.5 Notes and references...................................................316
Chapter 13. Serial rings and their properties.........................319
13.1 Finitely presented modules............................................ 319
13.2 The Drozd-Warfield theorem. Ore condition for serial rings............323
13.3 Minors of serial right Noetherian rings.................................325
13.4 Structure of serial right Noetherian rings...............................330
13.5 Serial right hereditary rings. Serial semiprime and right...................
Noetherian rings............................................................335
13.6 Notes and references...................................................339
Chapter 14. Semiperfect semidistributive rings .....................341
14.1 Distributive modules...................................................341
14.2 Reduction theorem for SPSD-rings....................................343
14.3 Quivers of SPSD-rings................................................345
14.4 Semiprime semiperfect rings...........................................347
14.5 Right Noetherian semiprime SPSD-rings..............................351
viii ALGEBRAS, RINGS AND MODULES
14.6 Quivers of tiled orders.................................................355
14.7 Quivers of exponent matrices..........................................357
14.8 Examples..............................................................361
14.9 Notes and references...................................................362
Suggestions for further reading........................................ 365
Index......................................................................369
Name index..............................................................377
Preface
Accosiativeringsandalgebrasareveryinterestingalgebraicstructures. Inastrict
sense, the theory of algebras (in particular, noncommutative algebras) originated
fromasingleexample,namelythequaternions,createdbySirWilliamR.Hamilton
in1843. Thiswasthefirstexampleofanoncommutative”numbersystem”. During
thenextfortyyearsmathematiciansintroducedotherexamplesofnoncommutative
algebras, began to bring some order into them and to single out certain types of
algebras for special attention. Thus, low-dimensional algebras, division algebras,
and commutative algebras, were classified and characterized. The first complete
results in the structure theory of associative algebras over the real and complex
fields were obtained by T.Molien, E.Cartan and G.Frobenius.
Modern ring theory began when J.H.Wedderburn proved his celebrated clas-
sification theorem for finite dimensional semisimple algebras over arbitrary fields.
Twenty years later, E.Artin proved a structure theorem for rings satisfying both
the ascending and descending chain condition which generalized Wedderburn
structure theorem. The Wedderburn-Artin theorem has since become a corner-
stone of noncommutative ring theory.
The purpose of this book is to introduce the subject of the structure theory of
associativerings. Thisbookisaddressedtoareaderwhowishestolearnthistopic
from the beginning to research level. We have tried to write a self-contained book
which is intended to be a modern textbook on the structure theory of associative
rings and related structures and will be accessible for independent study.
Thebasictoolsofinvestigationaremethodsfromthetheoryofmodules,which,
in our opinion, give a very simple and clear approach to both classical and new
results. Other interesting tools which we use for studying rings in this book are
techniquesfromthetheoryofquivers. Wedefinedifferentkindsofquiversofrings
and discuss various relations between the properties of rings and their quivers.
This is unusual and became possibly only recently, as the theory of quivers is a
quite new arrival in algebra.
Someofthetopicsofthebookhavebeenincludedbecauseoftheirfundamental
importance, others because of personal preference.
All rings considered in this book are associative with a nonzero identity.
The content of the book is divided into two volumes. The first volume is
devoted to both the standard classical theory of associative rings and to more
modern results of the theory of rings.
A large portion of the first volume of this book is based on the standard
university course in abstract and linear algebra and is fully accessible to students
in their second and third years. In particular, we do not assume knowledge of any
preliminary information on the theory of rings and modules.
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x ALGEBRAS, RINGS AND MODULES
A number of notes, some of them of a bibliographical others of a historical
nature, are collected at the end of each chapter.
In chapter 1 the fundamental tools for studying rings are introduced. In this
chapterwegiveanumberofbasicdefinitions,stateseveralfundamentalproperties
and give a number of different examples. Some important concepts that play a
central role in the theory of rings are introduced.
The main objects of chapter 2 are decomposition theorems for rings. In par-
ticular, much attention is given to the two-sided Peirce decomposition of rings.
In section 2.2. we study semisimple modules which form one of the most im-
portant classes of modules and play a distinguished role in the theory of modules.
ForsemisimpleringsweprovethefundamentalWedderburn-Artintheorem,which
givesthecompleteclassificationofsuchrings. Inthischapterthereisalsoprovided
a brief introduction to the theory of lattices and Boolean algebras. In section 2.4
we introduce finitely decomposable rings and finitely decomposable identity rings
andstudytheirmainproperties. Fortheseringsweprovethedecompositiontheo-
rems using the general theory of Boolean algebras and the theory of idempotents.
Chapter 3 is devoted to studying Noetherian and Artinian rings and modules.
In particular, we prove the famous Jordan-Ho¨lder theorem and the Hilbert basis
theorem. The most important part of this chapter is the study of the Jacobson
radicalanditsproperties. InthischapterwealsoproveNakayama’slemmawhich
is a simple result with powerful applications. Section 3.6 presents a criterion of
rings to be Noetherian or Artinian. In section 3.7 we consider semiprimary rings
and prove a famous theorem, due to Hopkins and Levitzki, which shows that any
Artinian ring is also Noetherian.
Chapter 4 presents the fundamental notions of the theory of homological al-
gebra, such as categories and functors. In particular, we introduce the functor
Hom and the tensor product functor and discuss the most important properties
of them. In this chapter we also study tensor product of modules and direct and
inverse limits.
Chapter 5 gives a brief study of special classes of modules, such as free, pro-
jective, injective, and flat modules. We also study hereditary and semihereditary
rings. Finally we consider the Herstein-Small rings, which provide an example of
rings which are right hereditary but not left hereditary.
Homological dimensions of rings and modules are discussed in chapter 6. In
this chapter derived functors and the functors Ext and Tor are introduced and
studied. This chapter presents the notions of projective and injective dimensions
of modules. We also define global dimensions of rings and give some principal
results of the theory of homological dimensions of rings.
In chapter 7 we consider different classes of commutative domains, such as
principal ideal domains, factorial rings and Euclidean domains. We study their
mainpropertiesandprovethefundamentalstructuretheoremforfinitelygenerated
modules over principal ideal domains. We also give the main applications of this
theorem to the study of finitely generated Abelian groups and canonical forms of
