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Tài liệu Advanced Modern Algebra by Joseph J. Rotman Hardcover: 1040 pages Publisher: Prentice
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Advanced Modern Algebra
by Joseph J. Rotman
Hardcover: 1040 pages
Publisher: Prentice Hall; 1st edition (2002); 2nd printing (2003)
Language: English
ISBN: 0130878685
Book Description
This book's organizing principle is the interplay between groups and rings,
where “rings” includes the ideas of modules. It contains basic definitions,
complete and clear theorems (the first with brief sketches of proofs), and
gives attention to the topics of algebraic geometry, computers, homology,
and representations. More than merely a succession of definition-theorem-proofs,
this text put results and ideas in context so that students can appreciate why
a certain topic is being studied, and where definitions originate. Chapter
topics include groups; commutative rings; modules; principal ideal domains;
algebras; cohomology and representations; and homological algebra. For
individuals interested in a self-study guide to learning advanced algebra and
its related topics.
Book Info
Contains basic definitions, complete and clear theorems, and gives attention
to the topics of algebraic geometry, computers, homology, and representations.
For individuals interested in a self-study guide to learning advanced algebra
and its related topics.
To my wife
Marganit
and our two wonderful kids,
Danny and Ella,
whom I love very much
Contents
Second Printing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Special Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter 1 Things Past ............................ 1
1.1. Some Number Theory ............................. 1
1.2. Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3. Some Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 2 Groups I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2. Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3. Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4. Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5. Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.6. Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7. Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Chapter 3 Commutative Rings I . . . . . . . . . . . . . . . . . . . . . 116
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.2. First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.4. Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.5. Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.6. Euclidean Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.7. Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.8. Quotient Rings and Finite Fields . . . . . . . . . . . . . . . . . . . . . . . 182
v
vi Contents
Chapter 4 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4.1. Insolvability of the Quintic . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Formulas and Solvability by Radicals . . . . . . . . . . . . . . . . . . . 206
Translation into Group Theory . . . . . . . . . . . . . . . . . . . . . . . 210
4.2. Fundamental Theorem of Galois Theory . . . . . . . . . . . . . . . . . . . 218
Chapter 5 Groups II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.1. Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
5.2. The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.3. The Jordan–Holder Theorem . ¨ . . . . . . . . . . . . . . . . . . . . . . . . 278
5.4. Projective Unimodular Groups . . . . . . . . . . . . . . . . . . . . . . . . 289
5.5. Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
5.6. The Nielsen–Schreier Theorem . . . . . . . . . . . . . . . . . . . . . . . . 311
Chapter 6 Commutative Rings II . . . . . . . . . . . . . . . . . . . . . 319
6.1. Prime Ideals and Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . . 319
6.2. Unique Factorization Domains . . . . . . . . . . . . . . . . . . . . . . . . 326
6.3. Noetherian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
6.4. Applications of Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 345
6.5. Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
6.6. Grobner Bases ¨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Generalized Division Algorithm . . . . . . . . . . . . . . . . . . . . . . 400
Buchberger’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Chapter 7 Modules and Categories . . . . . . . . . . . . . . . . . . . 423
7.1. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
7.2. Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
7.3. Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
7.4. Free Modules, Projectives, and Injectives . . . . . . . . . . . . . . . . . . . 471
7.5. Grothendieck Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
7.6. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
Chapter 8 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
8.1. Noncommutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
8.2. Chain Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
8.3. Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
8.4. Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
8.5. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
8.6. Theorems of Burnside and of Frobenius . . . . . . . . . . . . . . . . . . . 634
Contents vii
Chapter 9 Advanced Linear Algebra . . . . . . . . . . . . . . . . . . 646
9.1. Modules over PIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
9.2. Rational Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
9.3. Jordan Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
9.4. Smith Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
9.5. Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
9.6. Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
9.7. Division Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
9.8. Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
9.9. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
9.10. Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
Chapter 10 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
10.2. Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
10.3. General Extensions and Cohomology . . . . . . . . . . . . . . . . . . . . 794
10.4. Homology Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813
10.5. Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830
10.6. Ext and Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852
10.7. Cohomology of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 870
10.8. Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887
10.9. Introduction to Spectral Sequences . . . . . . . . . . . . . . . . . . . . . 893
Chapter 11 Commutative Rings III . . . . . . . . . . . . . . . . . . . 898
11.1. Local and Global . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898
11.2. Dedekind Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922
Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923
Nullstellensatz Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938
Characterizations of Dedekind Rings . . . . . . . . . . . . . . . . . . . . 948
Finitely Generated Modules over Dedekind Rings . . . . . . . . . . . . . 959
11.3. Global Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
11.4. Regular Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985
Appendix The Axiom of Choice and Zorn’s Lemma . . . . . . . . A-1
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1
Second Printing
It is my good fortune that several readers of the first printing this book apprised me of
errata I had not noticed, often giving suggestions for improvement. I give special thanks to
Nick Loehr, Robin Chapman, and David Leep for their generous such help.
Prentice Hall has allowed me to correct every error found; this second printing is surely
better than the first one.
Joseph Rotman
May 2003
viii
Preface
Algebra is used by virtually all mathematicians, be they analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. Nowadays, everyone agrees that some knowledge of linear algebra, groups, and commutative rings is
necessary, and these topics are introduced in undergraduate courses. We continue their
study.
This book can be used as a text for the first year of graduate algebra, but it is much more
than that. It can also serve more advanced graduate students wishing to learn topics on
their own; while not reaching the frontiers, the book does provide a sense of the successes
and methods arising in an area. Finally, this is a reference containing many of the standard
theorems and definitions that users of algebra need to know. Thus, the book is not only an
appetizer, but a hearty meal as well.
When I was a student, Birkhoff and Mac Lane’s A Survey of Modern Algebra was the
text for my first algebra course, and van der Waerden’s Modern Algebra was the text for
my second course. Both are excellent books (I have called this book Advanced Modern
Algebra in homage to them), but times have changed since their first appearance: Birkhoff
and Mac Lane’s book first appeared in 1941, and van der Waerden’s book first appeared
in 1930. There are today major directions that either did not exist over 60 years ago, or
that were not then recognized to be so important. These new directions involve algebraic
geometry, computers, homology, and representations (A Survey of Modern Algebra has
been rewritten as Mac Lane–Birkhoff, Algebra, Macmillan, New York, 1967, and this
version introduces categorical methods; category theory emerged from algebraic topology,
but was then used by Grothendieck to revolutionize algebraic geometry).
Let me now address readers and instructors who use the book as a text for a beginning
graduate course. If I could assume that everyone had already read my book, A First Course
in Abstract Algebra, then the prerequisites for this book would be plain. But this is not a
realistic assumption; different undergraduate courses introducing abstract algebra abound,
as do texts for these courses. For many, linear algebra concentrates on matrices and vector
spaces over the real numbers, with an emphasis on computing solutions of linear systems
of equations; other courses may treat vector spaces over arbitrary fields, as well as Jordan
and rational canonical forms. Some courses discuss the Sylow theorems; some do not;
some courses classify finite fields; some do not.
To accommodate readers having different backgrounds, the first three chapters contain
ix
x Preface
many familiar results, with many proofs merely sketched. The first chapter contains the
fundamental theorem of arithmetic, congruences, De Moivre’s theorem, roots of unity,
cyclotomic polynomials, and some standard notions of set theory, such as equivalence
relations and verification of the group axioms for symmetric groups. The next two chapters contain both familiar and unfamiliar material. “New” results, that is, results rarely
taught in a first course, have complete proofs, while proofs of “old” results are usually
sketched. In more detail, Chapter 2 is an introduction to group theory, reviewing permutations, Lagrange’s theorem, quotient groups, the isomorphism theorems, and groups acting
on sets. Chapter 3 is an introduction to commutative rings, reviewing domains, fraction
fields, polynomial rings in one variable, quotient rings, isomorphism theorems, irreducible
polynomials, finite fields, and some linear algebra over arbitrary fields. Readers may use
“older” portions of these chapters to refresh their memory of this material (and also to
see my notational choices); on the other hand, these chapters can also serve as a guide for
learning what may have been omitted from an earlier course (complete proofs can be found
in A First Course in Abstract Algebra). This format gives more freedom to an instructor,
for there is a variety of choices for the starting point of a course of lectures, depending
on what best fits the backgrounds of the students in a class. I expect that most instructors would begin a course somewhere in the middle of Chapter 2 and, afterwards, would
continue from some point in the middle of Chapter 3. Finally, this format is convenient
for the author, because it allows me to refer back to these earlier results in the midst of a
discussion or a proof. Proofs in subsequent chapters are complete and are not sketched.
I have tried to write clear and complete proofs, omitting only those parts that are truly
routine; thus, it is not necessary for an instructor to expound every detail in lectures, for
students should be able to read the text.
Here is a more detailed account of the later chapters of this book.
Chapter 4 discusses fields, beginning with an introduction to Galois theory, the interrelationship between rings and groups. We prove the insolvability of the general polynomial of degree 5, the fundamental theorem of Galois theory, and applications, such as a
proof of the fundamental theorem of algebra, and Galois’s theorem that a polynomial over
a field of characteristic 0 is solvable by radicals if and only if its Galois group is a solvable
group.
Chapter 5 covers finite abelian groups (basis theorem and fundamental theorem), the
Sylow theorems, Jordan–Holder theorem, solvable groups, simplicity of the linear groups ¨
PSL(2, k), free groups, presentations, and the Nielsen–Schreier theorem (subgroups of free
groups are free).
Chapter 6 introduces prime and maximal ideals in commutative rings; Gauss’s theorem
that R[x] is a UFD when R is a UFD; Hilbert’s basis theorem, applications of Zorn’s lemma
to commutative algebra (a proof of the equivalence of Zorn’s lemma and the axiom of
choice is in the appendix), inseparability, transcendence bases, Luroth’s theorem, affine va- ¨
rieties, including a proof of the Nullstellensatz for uncountable algebraically closed fields
(the full Nullstellensatz, for varieties over arbitrary algebraically closed fields, is proved
in Chapter 11); primary decomposition; Grobner bases. Chapters 5 and 6 overlap two ¨
chapters of A First Course in Abstract Algebra, but these chapters are not covered in most
Preface xi
undergraduate courses.
Chapter 7 introduces modules over commutative rings (essentially proving that all
R-modules and R-maps form an abelian category); categories and functors, including
products and coproducts, pullbacks and pushouts, Grothendieck groups, inverse and direct
limits, natural transformations; adjoint functors; free modules, projectives, and injectives.
Chapter 8 introduces noncommutative rings, proving Wedderburn’s theorem that finite
division rings are commutative, as well as the Wedderburn–Artin theorem classifying semisimple rings. Modules over noncommutative rings are discussed, along with tensor products, flat modules, and bilinear forms. We also introduce character theory, using it to prove
Burnside’s theorem that finite groups of order pmqn are solvable. We then introduce multiply transitive groups and Frobenius groups, and we prove that Frobenius kernels are normal
subgroups of Frobenius groups.
Chapter 9 considers finitely generated modules over PIDs (generalizing earlier theorems
about finite abelian groups), and then goes on to apply these results to rational, Jordan, and
Smith canonical forms for matrices over a field (the Smith normal form enables one to
compute elementary divisors of a matrix). We also classify projective, injective, and flat
modules over PIDs. A discussion of graded k-algebras, for k a commutative ring, leads to
tensor algebras, central simple algebras and the Brauer group, exterior algebra (including
Grassmann algebras and the binomial theorem), determinants, differential forms, and an
introduction to Lie algebras.
Chapter 10 introduces homological methods, beginning with semidirect products and
the extension problem for groups. We then present Schreier’s solution of the extension
problem using factor sets, culminating in the Schur–Zassenhaus lemma. This is followed
by axioms characterizing Tor and Ext (existence of these functors is proved with derived
functors), some cohomology of groups, a bit of crossed product algebras, and an introduction to spectral sequences.
Chapter 11 returns to commutative rings, discussing localization, integral extensions,
the general Nullstellensatz (using Jacobson rings), Dedekind rings, homological dimensions, the theorem of Serre characterizing regular local rings as those noetherian local
rings of finite global dimension, the theorem of Auslander and Buchsbaum that regular
local rings are UFDs.
Each generation should survey algebra to make it serve the present time.
It is a pleasure to thank the following mathematicians whose suggestions have greatly
improved my original manuscript: Ross Abraham, Michael Barr, Daniel Bump, Heng Huat
Chan, Ulrich Daepp, Boris A. Datskovsky, Keith Dennis, Vlastimil Dlab, Sankar Dutta,
David Eisenbud, E. Graham Evans, Jr., Daniel Flath, Jeremy J. Gray, Daniel Grayson,
Phillip Griffith, William Haboush, Robin Hartshorne, Craig Huneke, Gerald J. Janusz,
David Joyner, Carl Jockusch, David Leep, Marcin Mazur, Leon McCulloh, Emma Previato,
Eric Sommers, Stephen V. Ullom, Paul Vojta, William C. Waterhouse, and Richard Weiss.
Joseph Rotman
Etymology
The heading etymology in the index points the reader to derivations of certain mathematical
terms. For the origins of other mathematical terms, we refer the reader to my books Journey
into Mathematics and A First Course in Abstract Algebra, which contain etymologies of
the following terms.
Journey into Mathematics:
π, algebra, algorithm, arithmetic, completing the square, cosine, geometry, irrational
number, isoperimetric, mathematics, perimeter, polar decomposition, root, scalar, secant,
sine, tangent, trigonometry.
A First Course in Abstract Algebra:
affine, binomial, coefficient, coordinates, corollary, degree, factor, factorial, group,
induction, Latin square, lemma, matrix, modulo, orthogonal, polynomial, quasicyclic,
September, stochastic, theorem, translation.
xii
Special Notation
A algebraic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
An alternating group on n letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Ab category of abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
Aff(1, k) one-dimensional affine group over a field k . . . . . . . . . . . . . . . . . . . . . 125
Aut(G) automorphism group of a group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Br(k),Br(E/k) Brauer group, relative Brauer group ....................... 737, 739
C complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
C•, (C•, d•) complex with differentiations dn : Cn → Cn−1 . . . . . . . . . . . . . . . . . . 815
CG(x) centralizer of an element x in a group G . . . . . . . . . . . . . . . . . . . . . . . 101
D(R) global dimension of a commutative ring R . . . . . . . . . . . . . . . . . . . . . 974
D2n dihedral group of order 2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
deg( f ) degree of a polynomial f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Deg( f ) multidegree of a polynomial f (x1,..., xn) . . . . . . . . . . . . . . . . . . . . . 402
det(A) determinant of a matrix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757
dimk (V) dimension of a vector space V over a field k . . . . . . . . . . . . . . . . . . . . 167
dim(R) Krull dimension of a commutative ring R . . . . . . . . . . . . . . . . . . . . . . 988
Endk (M) endomorphism ring of a k-module M . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Fq finite field having q elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Frac(R) fraction field of a domain R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Gal(E/k) Galois group of a field extension E/k . . . . . . . . . . . . . . . . . . . . . . . . . 200
GL(V) automorphisms of a vector space V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
GL(n, k) n × n nonsingular matrices, entries in a field k . . . . . . . . . . . . . . . . . . 179
H division ring of real quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
Hn, Hn homology, cohomology . . ................................ 818, 845
ht(p) height of prime ideal p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987
Im integers modulo m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
I or
√
In identity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
I radical of an ideal I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Id(A) ideal of a subset A ⊆ kn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
im f image of a function f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
irr(α, k) minimal polynomial of α over a field k . . . . . . . . . . . . . . . . . . . . . . . . . 189
xiii
xiv Special Notation
k algebraic closure of a field k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
K0(R), K0(C) Grothendieck groups, direct sums . ........................ 491, 489
K
(C) Grothendieck group, short exact sequences . . . . . . . . . . . . . . . . . . . . . 492
ker f kernel of a homomorphism f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
l D(R) left global dimension of a ring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974
Matn(k) ring of all n × n matrices with entries in k . . . . . . . . . . . . . . . . . . . . . . 520
RMod category of left R-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
ModR category of right R-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
N natural numbers = {integers n : n ≥ 0} ........................... 1
NG(H) normalizer of a subgroup H in a group G . . . . . . . . . . . . . . . . . . . . . . 101
OE ring of integers in an algebraic number field E . . . . . . . . . . . . . . . . . . 925
O(x) orbit of an element x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
PSL(n, k) projective unimodular group = SL(n, k)/center . . . . . . . . . . . . . . . . . 292
Q rational numbers
Q quaternion group of order 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Qn generalized quaternion group of order 2n . . . . . . . . . . . . . . . . . . . . . . . 298
R real numbers
Sn symmetric group on n letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
SX symmetric group on a set X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
sgn(α) signum of a permutation α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
SL(n, k) n × n matrices of determinant 1, entries in a field k . . . . . . . . . . . . . . . 72
Spec(R) the set of all prime ideals in a commutative ring R . . . . . . . . . . . . . . 398
U(R) group of units in a ring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
UT(n, k) unitriangular n × n matrices over a field k . . . . . . . . . . . . . . . . . . . . . . 274
T I3 I4, a nonabelian group of order 12 . . . . . . . . . . . . . . . . . . . . . . . . 792
tG torsion subgroup of an abelian group G . . . . . . . . . . . . . . . . . . . . . . . . 267
tr(A) trace of a matrix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
V four-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Var(I) variety of an ideal I ⊆ k[x1,..., xn] . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Z integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Zp p-adic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Z(G) center of a group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Z(R) center of a ring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
[G : H] index of a subgroup H ≤ G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
[E : k] degree of a field extension E/k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
S T coproduct of objects in a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
S T product of objects in a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
S ⊕ T external, internal direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
K × Q direct product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
K
Q semidirect product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790
Ai direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Ai direct product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Special Notation xv
lim
←− Ai inverse limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
lim
−→ Ai direct limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
G commutator subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Gx stabilizer of an element x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
G[m] {g ∈ G : mg = 0}, where G is an additive abelian group . . . . . . . . . 267
mG {mg : g ∈ G}, where G is an additive abelian group . . . . . . . . . . . . . 253
G p p-primary component of an abelian group G . . . . . . . . . . . . . . . . . . . 256
k[x] polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
k(x) rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
k[[x]] formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
kX polynomials in noncommuting variables . . . . . . . . . . . . . . . . . . . . . . . 724
Rop opposite ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
Ra or (a) principal ideal generated by a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
R× nonzero elements in a ring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
H ≤ G H is a subgroup of a group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
H < G H is a proper subgroup of a group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
H ✁ G H is a normal subgroup of a group G . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A ⊆ B A is a submodule (subring) of a module (ring)B . . . . . . . . . . . . . . . . 119
A B A is a proper submodule (subring) of a module (ring)B . . . . . . . . . . 119
1X identity function on a set X
1X identity morphism on an object X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
f : a → b f (a) = b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
|X| number of elements in a set X
Y [T ]X matrix of a linear transformation T relative to bases X and Y . . . . . 173
χσ character afforded by a representation σ . . . . . . . . . . . . . . . . . . . . . . . . 610
φ(n) Euler φ-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
n
r
binomial coefficient n!/r!(n − r)! ............................... 5
δi j Kronecker delta δi j =
1 if i = j;
0 if i = j.
a1,..., ai,..., an list a1,..., an with ai omitted