Foundations of Commutative Rings and Their Modules

711 Pages·2016·4.34 MB·English

by  Fanggui Wang, Hwankoo Kim

Algebra and Applications
Fanggui Wang
Hwankoo Kim
Foundations of
Commutative
Rings and Their
Modules
Foundations of Commutative Rings
and Their Modules
Algebra and Applications
Volume 22
Series editors:
Michel Broué,
Université Paris Diderot, Paris, France
Alice Fialowski,
Eötvös Loránd University, Budapest, Hungary
Eric Friedlander,
University of Southern California, Los Angeles, USA
Iain Gordon,
University of Edinburgh, Edinburgh, UK
John Greenless,
Sheffield University, Sheffield, UK
Gerhard Hiß,
Aachen University, Aachen, Germany
Ieke Moerdijk,
Radboud University Nijmegen, Nijmegen, The Netherlands
Christoph Schweigert,
Hamburg University, Hamburg, Germany
Mina Teicher,
Bar-llan University, Ramat-Gan, Israel
Alain Verschoren,
University of Antwerp, Antwerp, Belgium
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Fanggui Wang Hwankoo Kim
(cid:129)
Foundations of Commutative
Rings and Their Modules
123
FangguiWang HwankooKim
Schoolof Mathematics Schoolof Computer andInformation
SichuanNormal University Engineering
Chengdu Hoseo University
China Asan
Korea (Republicof)
ISSN 1572-5553 ISSN 2192-2950 (electronic)
Algebra andApplications
ISBN978-981-10-3336-0 ISBN978-981-10-3337-7 (eBook)
DOI10.1007/978-981-10-3337-7
LibraryofCongressControlNumber:2016960184
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Preface
There are different approaches to characterizing the structures of commutative
rings:thecategoryofmodulesovercommutativerings,homologytheories,theories
ofthestaroperationonintegral domains,thegeneral theoryofcommutativerings,
andrelativehomologytheories.Thesehavedifferentemphases;however,theyhave
some common ground: the basic theory of commutative rings. This book starts by
elaborating this theory. Thus, this book is intended to serve as a textbook for a
course in commutative algebra at a graduate level and as a reference book for
researchers.
Aglanceatthecontentsofthefirstfivechaptersshowsthatwecoverthosetopics
normally included in any commutative algebra text, although to a greater level of
detailthanotherbooks.However,thecontentsinthisbook’sdiffersignificantlyfrom
themostcommutativealgebratexts,namelyourtreatmentoftheDedekind–Mertens
formula(Sect.1.7),the(small)finitisticdimensionofaring(Sect.3.10),Gorenstein
rings (Sect. 4.6), valuation overrings and the valuative dimension (Sect. 5.4), and
Nagata rings as quotient rings of polynomial rings (Sect. 5.5). On the other hand,
Chap.6presentsw-modulesovercommutativeringsastheycanbemostcommonly
usedbytorsiontheoryandmultiplicativeidealtheorysincethew-operationtheoryis
abridgecloselyconnectingtorsiontheorywithmultiplicativeidealtheory.Chapter7
dealswithmultiplicativeidealtheoryoverintegraldomains,whichcanbethoughtof
generalizationsandextensionsofworkdonebyR.GilmerinthebookMultiplicative
Ideal Theory [68]. In Chap. 8, we collect various results of the pullbacks, more
specially Milnor squares and D þ M constructions, which are probably the most
important example generating machine. In Chap. 9, we probe coherent rings with
finite weak global dimension and try to elaborate on the local ring of weak global
dimension two by combining homological tricks and methods of star operation
theoryintroducedinChap.7.Chapter10isdevotedtotheGrothendieckgroupofa
commutative ring. In particular, the Bass–Quillen problem is discussed. Finally,
Chap. 11 aims to introduce relative homological algebra, especially where the
related concepts of integral domains which appear in classical ideal theory are
defined and investigated by using the class of Gorenstein projective modules.
Inordertokeepthisbookfrombecomingtoounwieldy,weomittedimportanttopics
vii
viii Preface
such as generalizations of class groups and Kronecker function rings and some
generalizations of Krull domains.
While at least portions of the first five chapters should be read in order, the
remainingchaptersareessentiallyindependentofeachother,exceptforChaps.6–8.
Thosesectionsthatareessentiallyapplicationsofpreviousconceptsorelsearenot
necessary for the rest of the book.
Each section of this book is followed by a selection of exercises, of varying
degreesofdifficulty.Theexercisesshoulddeepenthereader’sunderstandingofthe
conceptspresentedinthisbook,althoughsomemaybelimitedtothelengthofthe
supplement related to content.
Chengdu, China Fanggui Wang
Asan, Korea (Republic of) Hwankoo Kim
May 2016
Contents
1 Basic Theory of Rings and Modules .... .... .... .... ..... .... 1
1.1 Basic Concepts of Rings and Modules... .... .... ..... .... 1
1.1.1 Rings and Ideals . .... .... .... .... .... ..... .... 1
1.1.2 Basic Concepts of Modules. .... .... .... ..... .... 3
1.1.3 Direct Product of Rings, Direct Product and Direct
Sum of Modules . .... .... .... .... .... ..... .... 6
1.2 Ring Homomorphisms and Module Homomorphisms..... .... 7
1.2.1 Ring Homomorphisms. .... .... .... .... ..... .... 7
1.2.2 Module Homomorphisms .. .... .... .... ..... .... 10
1.3 Finitely Generated Modules and Matrix Methods... ..... .... 14
1.3.1 Finitely Generated Modules. .... .... .... ..... .... 14
1.3.2 Simple Modules, Maximal Submodules,
and Zorn’s Lemma ... .... .... .... .... ..... .... 16
1.3.3 Jacobson Radical of a Ring. .... .... .... ..... .... 18
1.3.4 Matrix Methods.. .... .... .... .... .... ..... .... 18
1.4 Prime Ideals and Nil Radical .. .... .... .... .... ..... .... 22
1.4.1 Prime Ideals..... .... .... .... .... .... ..... .... 22
1.4.2 Nil Radical and Radical of an Ideal... .... ..... .... 24
1.5 Quotient Rings and Quotient Modules ... .... .... ..... .... 25
1.5.1 Local Rings..... .... .... .... .... .... ..... .... 25
1.5.2 Quotient Rings .. .... .... .... .... .... ..... .... 26
1.5.3 Quotient Modules .... .... .... .... .... ..... .... 28
1.6 Free Modules, Torsion Modules, and Torsion-Free Modules ... 33
1.6.1 Free Modules.... .... .... .... .... .... ..... .... 33
1.6.2 Torsion Modules, Torsion-Free Modules,
and Divisible Modules. .... .... .... .... ..... .... 36
1.7 Polynomial Rings and Power Series Rings.... .... ..... .... 37
1.7.1 Polynomial Rings over One Indeterminate . ..... .... 37
1.7.2 Polynomials with Coefficients in a Module. ..... .... 41
ix
x Contents
1.7.3 Dedekind–Mertens Formula. .... .... .... ..... .... 42
1.7.4 Polynomial Rings over Many Indeterminates
and Formal Power Series Rings
over One Indeterminate.... .... .... .... ..... .... 45
1.8 Krull Dimension of a Ring.... .... .... .... .... ..... .... 47
1.8.1 Basic Properties of Krull Dimension of a Ring... .... 47
1.8.2 Krull Dimension of a Polynomial Ring.... ..... .... 48
1.8.3 Connected Rings. .... .... .... .... .... ..... .... 50
1.9 Exact Sequences and Commutative Diagrams.. .... ..... .... 51
1.9.1 Exact Sequences . .... .... .... .... .... ..... .... 52
1.9.2 Five Lemma and Snake Lemma . .... .... ..... .... 53
1.9.3 Completion of Diagrams... .... .... .... ..... .... 59
1.9.4 Pushout and Pullback . .... .... .... .... ..... .... 61
1.10 Exercises. .... .... ..... .... .... .... .... .... ..... .... 63
2 The Category of Modules..... .... .... .... .... .... ..... .... 71
2.1 The Functor Hom.. ..... .... .... .... .... .... ..... .... 71
2.1.1 Categories. ..... .... .... .... .... .... ..... .... 71
2.1.2 Functors... ..... .... .... .... .... .... ..... .... 74
2.1.3 Basic Properties of the Functor Hom.. .... ..... .... 75
2.1.4 Natural Transforms of Functors.. .... .... ..... .... 77
2.1.5 Torsionless Modules and Reflexive Modules..... .... 78
2.2 The Functor (cid:2) .... ..... .... .... .... .... .... ..... .... 80
2.2.1 Bilinear Mappings and Tensor Products ... ..... .... 80
2.2.2 Basic Properties of the Functor (cid:2).... .... ..... .... 81
2.2.3 Change of Rings and Adjoint
Isomorphism Theorem. .... .... .... .... ..... .... 83
2.2.4 Tensor Product and Localization. .... .... ..... .... 85
2.3 Projective Modules. ..... .... .... .... .... .... ..... .... 86
2.3.1 Projective Modules ... .... .... .... .... ..... .... 87
2.3.2 Kaplansky Theorem... .... .... .... .... ..... .... 90
2.4 Injective Modules.. ..... .... .... .... .... .... ..... .... 93
2.4.1 Injective Modules .... .... .... .... .... ..... .... 93
2.4.2 Injective Envelope of a Module.. .... .... ..... .... 97
2.5 Flat Modules.. .... ..... .... .... .... .... .... ..... .... 100
2.5.1 Flat Modules and Their Characterizations.. ..... .... 100
2.5.2 Faithfully Flat Modules.... .... .... .... ..... .... 106
2.5.3 Direct Limits.... .... .... .... .... .... ..... .... 109
2.6 Finitely Presented Modules.... .... .... .... .... ..... .... 114
2.6.1 Finitely Presented Modules. .... .... .... ..... .... 114
2.6.2 Isomorphism Theorems Related to Hom and (cid:2)... .... 119