Download Free eBook on Abstract Algebra

488 Pages·2012·2.01 MB·English

Andreas Bernhard Zeidler
Abstract Algebra
Rings, Modules, Polynomials, Ring Extensions,
Categorical and Commutative Algebra
February 15, 2012 (488 pages)
If you have read this text I would like to invite you to contribute to it:
Comments, corrections and suggestions are very much appreciated, at
[email protected], or visit my homepage at
www.mathematik.uni-tuebingen.de/ab/algebra/index.html
This book is dedicated to the entire mathematical society.
To all those who contribute to mathematics and keep it alive by teaching it.
Contents
0 Prelude 5
0.1 About this Book . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.2 Notation and Symbols . . . . . . . . . . . . . . . . . . . . . . 9
0.3 Mathematicians at a Glance . . . . . . . . . . . . . . . . . . . 15
1 Groups and Rings 19
1.1 Defining Groups . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3 Defining Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5 First Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.6 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.8 Ordered Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 72
1.9 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2 Commutative Rings 83
2.1 Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.2 Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.3 Radical Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.4 Noetherian Rings . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.5 Unique Factorisation Domains. . . . . . . . . . . . . . . . . . 103
2.6 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 114
2.7 Lasker-Noether Decomposition . . . . . . . . . . . . . . . . . 124
2.8 Finite Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
2.9 Localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.10 Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.11 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . 153
3 Modules 159
3.1 Defining Modules . . . . . . . . . . . . . . . . . . . . . . . . . 159
3.2 First Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.3 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.4 Ideal-induced Modules . . . . . . . . . . . . . . . . . . . . . . 182
3.5 Block Decompositions . . . . . . . . . . . . . . . . . . . . . . 186
3.6 Dependence Relations . . . . . . . . . . . . . . . . . . . . . . 187
3.7 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . 191
3.8 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.9 Isomorphism Theorems . . . . . . . . . . . . . . . . . . . . . 203
3.10 Rank of Modules . . . . . . . . . . . . . . . . . . . . . . . . . 207
3.11 Noetherian Modules . . . . . . . . . . . . . . . . . . . . . . . 213
3.12 Localisation of Modules . . . . . . . . . . . . . . . . . . . . . 221
4 Linear Algebra 224
4.1 Matix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.2 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . 232
4.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 238
4.4 Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . 252
4.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
4.6 Rank of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 252
4.7 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . 252
5 Spectral Theory 253
5.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.3 Operatoralgebras . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.4 Diagonalisation . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.5 Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 253
6 Structure Theorems 254
6.1 Associated Primes . . . . . . . . . . . . . . . . . . . . . . . . 254
6.2 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . 254
6.3 The Theorem of Pru¨fer. . . . . . . . . . . . . . . . . . . . . . 254
6.4 Length of Modules . . . . . . . . . . . . . . . . . . . . . . . . 254
7 Polynomial Rings 255
7.1 Monomial Orders . . . . . . . . . . . . . . . . . . . . . . . . . 255
7.2 Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.3 Defining Polynomials . . . . . . . . . . . . . . . . . . . . . . . 268
7.4 The Standard Cases . . . . . . . . . . . . . . . . . . . . . . . 268
7.5 Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . 268
7.6 Derivation of Polynomials . . . . . . . . . . . . . . . . . . . . 268
7.7 Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . 268
7.8 Gr¨obner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 268
7.9 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
8 Polynomials in One Variable 269
8.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.2 Irreducibility Tests . . . . . . . . . . . . . . . . . . . . . . . . 269
8.3 Symmetric Polynomials . . . . . . . . . . . . . . . . . . . . . 269
8.4 The Resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.5 The Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.6 Polynomials of Low Degree . . . . . . . . . . . . . . . . . . . 269
8.7 Polynomials of High Degree . . . . . . . . . . . . . . . . . . . 269
8.8 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . 269
9 Group Theory 270
10 Multilinear Algebra 276
10.1 Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . 276
10.2 Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 276
10.3 Tensor Product of Modules . . . . . . . . . . . . . . . . . . . 276
10.4 Tensor Product of Algebras . . . . . . . . . . . . . . . . . . . 276
10.5 Tensor Product of Maps . . . . . . . . . . . . . . . . . . . . . 276
10.6 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
11 Categorial Algebra 277
11.1 Sets and Classes . . . . . . . . . . . . . . . . . . . . . . . . . 277
11.2 Categories and Functors . . . . . . . . . . . . . . . . . . . . . 277
11.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11.4 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11.5 Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . 278
11.6 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
12 Ring Extensions 279
13 Galois Theory 280
14 Graded Rings 281
15 Valuations 282
16 Proofs - Fundamentals 284
17 Proofs - Rings and Modules 306
18 Proofs - Commutative Algebra 413
Chapter 0
Prelude
0.1 About this Book
The Aim of this Book
Mathematics knows two directions - analysis and algebra - and any math-
ematical discipline can be weighted how analytical resp. algebraical it is.
Analysis is characterized by having a notion of convergence that allows to
approximate solutions (and reach them in the limit). Algebra is character-
ized by having no convergence and hence allowing finite computations only.
This book now is meant to be a thorough introduction into algebra.
Likewise every textbook on mathematics is drawn between two pairs of
extremes: (easy understandability versus great generality) and (complete-
ness versus having a clear line of thought). Among these contrary poles we
usually chose (generality over understandability) and (completeness over a
clear red line). Nevertheless we try to reach understandability by beingvery
precise and accurate and including many remarks and examples.
At last some personal philosophy: a perfect proof is like a perfect gem -
unbreakably hard, spotlessly clear, flawlessly cut and beautifully displayed.
In this book we are trying to collect such gemstones. And we are proud to
claim, that we are honest about where a proof is due and present complete
proofs of almost every claim contained herein (which makes this textbook
very different from most others).
This Book is Written for
many different kinds of mathematicians: primarily is meant to be a source
of reference for intermediate to advanced students, who have already had
a first contact with algebra and now closely examine some topic for their
seminars, lectures or own thesis. But because of its great generality and
completeness it is also suited as an encyclopaedia for professors who pre-
pare their lectures and researchers who need to estimate how far a certain
method carries. Frankly this book is not perfectly suited to be a monograph
fornovicestomathematics. Soifyouareonewethinkyoucangreatlyprofit
fromthisbook,butyouwillprobablyhavetoconsultadditionalmonographs
(at a more introductory level) to help you understand this text.
5
Prerequisites
We take for granted, that the reader is familiar with the basic notions of
naivelogic(statements, implication,proofbycontradiction, usageofquanti-
fiers, ...) and naive set theory (Cantor’s notion of a set, functions, partially
ordered sets, equivalence relations, Zorn’s Lemma, ...). We will present a
short introduction to classes and the NBG axioms when it comes to cate-
gory theory. Further we require some basic knowledge of integers (including
proofs by induction) and how to express them in decimal numbers. We will
sometimes use the field of real numbers, as they are most probably well-
known to the reader, but they are not required to understand this text.
Aside from these prerequisites we will start from scratch.
Topics Covered
Westartbyintroducinggroupsandrings, immediatelyspecializingonrings.
Of general ring theory we will introduce the basic notions only, e.g. the
isomorphism theorems. Then we will turn our attention to commutative
rings, which will be the first major topic of this book: we closely study
maximal ideals, prime ideals, intersections of such (radical ideals) and the
relations to localisation. Further we will study rings with chain conditions
(noetherian and artinian rings) including the Lasker-Noether theorem. This
willleadtostandardtopicslikethefundamentaltheoremofarithmetic. And
we conclude commutative ring theory by studying discrete valuation rings,
Dedekind domains and Krull rings.
Then we will turn our attention to modules, including rank, dimension
and length. We will see that modules are a natural and powerful general-
isation of ideals and large parts of ring theory generalises to this setting,
e.g. localisation and primary decomposition. Module theory naturally leads
to linear algebra, i.e. the theory of matrix representations of a homomor-
phism of modules. Applying the structure theorems of modules (the theo-
rem of Pru¨fer to be precise) we will treat canonical form theory (e.g. Jordan
normal form).
Next we will study polynomials from top down: that is we introduce
general polynomial rings (also known as group algebras) and graded alge-
bras. Only then we will regard the more classical problems of polynomials
in one variable and their solvability. Finally we will regard polynomials in
several variables again. Using Gr¨obner bases it is possible to solve abstract
algebraic questions by purely computational means.
Then we will return to group theory: most textbooks begin with this
topic,butwechosenotto. Eventhoughgrouptheoryseemstobeelementary
andfundamentalthisisnotquitetrue. Infactitheavilyreliesonarguments
like divisibility and prime decomposition in the integers, topics that are
native to ring theory. And commutative groups are best treated from the
point of view of module theory. Never the less you might as well skip the
previous sections and start with group theory right away. We will present
the standard topics: the isomorphism theorems, group actions (including
the formula of Burnside), the theorems of Sylow and lastly the p-q-theorem.
Howeverweareaimingdirectlyfortherepresentationtheoryoffinitegroups.
Thefirstpartisconcludedbypresentingathoroughintroductiontowhat
is called multi-linear algebra. We will study dual pairings, tensor products
of modules (and algebras) over a commutative base ring, derivations and
the module of differentials.
6 0 Prelude
Thuswehavegatheredawholebunchofseparatetheories-anditistimefor
a second structurisation (the first structurisation being algebra itself). We
will introduce the notions of categories, functors, equivalence of categories,
(co-)products and so on. Categories are merely a manner of speaking -
nothingthatcanbedonewithcategorytheorycouldnothavebeenachieved
without. Yet the language of categories presents a unifying concept for all
the different branches of mathematics, extending far beyond algebra. So we
will first recollect which part of the theory we have established is what in
the categorical setting. The categorial language is the right setting to treat
chain complexes, especially exact sequences of modules. Finally we will
present the basics of abelian categories as a unifying concept of all those
separate theories.
We will then aim for some more specialised topics: At first we will study
ring extensions and the dimension theory of commutative rings. A special
casearefieldextensionsincludingthebeautifultopicofGaloistheory. After-
wards we turn our attention to filtrations, completions, zeta-functions and
the Hilbert-Samuel polynomial. Finally we will venture deeper into number
theory: studying the theory of valuations up to the theorem of Riemann-
Roche for number fields.
Topics not Covered
There are many instances where, dropping a finiteness condition, one has to
introduce some topology in order to pursue the theory further. Examples
are: linear algebra on infinite dimensional vector-spaces, representation the-
ory of infinite groups and Galois theory of infinite field extensions. Another
natural extension would be to introduce the Zariski topology on the spec-
trum of a ring, which would lead to the theory of schemes directly. Yet the
scope of this text is purely algebraic and hence we will usually stop at the
point where topology sets in (but give hints for further readings).
The Two Parts
Mathematics has a peculiarity to it: there are problems (and answers) that
are easy to understand but hard to prove. The most famous example is
Fermat’s Last Theorem - the statement (for any n ≥ 3 there are no non-
trivial integers (a,b,c) ∈ Z3 that satisfy the equation an + bn = cn) can
be understood by anyone. Yet the proof is extremely hard to provide. Of
course this theorem has no value of its own (it is the proof that contains
deep insights into the structure of mathematics), but this is no general rule.
E.g.thetheoremofWedderburn(everyfiniteskew-fieldisafield)iseasyand
useful, but its proof will be performed using a beautiful trick-computation
(requiring the more advanced method of cyclotomic polynomials).
Thus we have chosen an unusual approach: we have separated the truth
(i.e. definitions, examples and theorems) from their proofs. This enables us
to present the truth in a stringent way, that allows the reader to get a feel
for the mathematical objects displayed. Most of the proofs could have been
given right away, but in several cases the proof of a statement can only be
done after we have developed the theory further. Thus the sequel of theo-
rems may (and will) be different from the order in which they are proved.
Hence the two parts.
0.1 About this Book 7
Our Best Advice
It is a well-known fact, that some proofs are just computational and only
contain little (or even no) insight into the structure of mathematics. Others
are brilliant, outstanding insights that are of no lesser importance than the
theorem itself. Thus we have already included remarks of how the proof
works in the first part of this book. And our best advice is to read a section
entirely to get a feel for the objects involved - only then have a look at the
proofs that have been recommended. Ignore the other proofs, unless you
have to know about them, for some reason. At several occasions this text
contains the symbols (♦) and ((cid:4)) . These are meant to guide the reader in
the following ways:
(♦) As we have assorted the topics covered thematically (paying little at-
tentiontothesequelofproofs)itmighthappenthatacertainexample
or theorem is far beyond the scope of the theory presented so far. In
this case the reader is asked to read over it lightly (or even skip it
entirely) and return to it later (after he has gained some more experi-
ence).
((cid:4)) On some very rare occasions we will append a theorem without giving
aproof(iftheproofisbeyondthescopeofthistext). Suchaninstance
will be marked by the black box symbol. In this case we will always
give a complete reference of the most readable proof the author is
aware of. And this symbol will be hereditarily, that is once we use
a theorem that has not been proved any other proof relying on the
unproved statement will also be branded by the black box symbol.
8 0 Prelude
0.2 Notation and Symbols
Conventions
We now wish to include a set of the frequently used symbols, conventions
and notations. In particular we clarify the several domains of numbers.
• First of all we employ the nice convention (introduced by Halmos) to
write iff as an abbreviation for if and only if.
• We denote the set of natural numbers - i.e. the positive integers in-
cluding zero - by N := {0,1,2,3,...}. Further for any two integers
a,b ∈ Z we denote the interval of integer numbers ranging from a to
b by a...b := {k ∈ Z | a ≤ k ≤ b}.
• We will denote the set of integers by Z = N∪(−N), and the rationals
by Q = {a/b | a,b ∈ Z,b (cid:54)= 0}. Whereas Z will be taken for granted,
Q will be introduced as the quotient field of Z.
• The reals will be denoted by R and we will present an example of how
they can be defined (without proving their properties however). The
complex numbers will be denoted by C = {a+ib | a,b ∈ R} and we
will present several ways of constructing them.
• (♦) We will sometimes use the Kronecker-Symbol δ(a,b) (in the liter-
ature this is also denoted by δ ), which is defined to be
a,b
(cid:26)
1 if a = b
δ(a,b) = δ :=
a,b 0 if a (cid:54)= b
Inmostcasesaandb ∈ Zwillbeintegersand0,1 ∈ Zwillbeintegers,
too. Butingeneralwearegivensomering(R,+,·)anda,b ∈ R. Then
the elements 0 and 1 ∈ R on the right hand side are taken to be the
zero-element 0 and unit-element 1 of R again.
• We will write A ⊆ X to indicate that A is a subset of X and A ⊂ X
will denote strict inclusion (i.e. A ⊆ X and there is some x ∈ X with
x (cid:54)∈ A). For any set X we denote its power set (i.e. the set of all
its subsets) by P(X) := {A | A ⊆ X}. And for a subset A ⊆ X we
denote the complement of A in X by CA := X \A.
• Listing several elements x ,...,x ∈ X of some set X, we do not re-
1 n
quirethesex tobepairwisedistinct(e.g.x = x mightwellhappen).
i 1 2
Yet if we only give explicit names x to the elements of some previ-
i
ously given subset A = {x ,...,x } ⊆ X we already consider the x
1 n i
to be pairwise distinct (that is x = x implies i = j). Note that if the
i j
x (not the set {x ,...,x }) have been given, then {x ,...,x } may
i 1 n 1 n
hence contain fewer than n elements!
(cid:83)
• Given an arbitrary set of sets A one defines the grand union A and
(cid:84)
the grand intersection A to be the set consisting of all elements a
that are contained in one (resp. all) of the sets A ∈ A, formally
(cid:91)
A := {a | ∃A ∈ A : a ∈ A}
(cid:92)
A := {a | ∀A ∈ A : a ∈ A}
0.2 Notation and Symbols 9
(cid:84)
Notethat Aonlyisawell-definedset,ifA =(cid:54) ∅isnon-empty. Awell-
knownspecialcaseofthisisthefollowing: consideranytwosetsAand
(cid:83) (cid:84)
B and let A := {A,B}. Then A∪B = A and A∩B = A. This
notion is just a generalisation of the ordinary union and intersection
to arbitrary collections A of sets.
• If X and Y are any sets then we will denote the set of all functions
from X to Y by F(X,Y) = YX = {f | f : X → Y }. And for any
such function f : X → Y : x (cid:55)→ f(x) we will denote its graph (note
that from the set-theoretical point of view f is its graph) by
Γ(f) := {(x,f(x)) | x ∈ X} ⊆ X ×Y
• Oncewehavedefinedfunctions,itiseasytodefinearbitrarycartesian
products. That is let I (cid:54)= ∅ be any non-empty set and for any i ∈ I
let X be another set. Let us denote the union of all the X by X
i i
(cid:91)
X := X = {x | ∃i ∈ I : x ∈ X }
i i
i∈I
Then the cartesian product of the X consists of all the functions
i
x : I → X such that for any i ∈ I we have x := x(i) ∈ X . Note that
i i
thereby it is customary to write (x ) in place of x. Formally
i
(cid:89)
X := {x : I → X : i (cid:55)→ x | ∀i ∈ I : x ∈ X }
i i i i
i∈I
• If I (cid:54)= ∅ is any index set and A ⊆ X is a non-empty A (cid:54)= ∅ subset
i i
of X (where i ∈ I) then the axiom of choice states that there is
a function a : I → X such that for any i ∈ I we get a(i) ∈ A . In
i
other words the product of non-empty sets is non-empty again. It
is a remarkable fact that this seemingly trivial property has far-flung
consequences, the lemma of Zorn being the most remarkable one.
(cid:89)
X (cid:54)= ∅ ⇐⇒ ∀ i ∈ I : X (cid:54)= ∅
i i
i∈I
• Let X (cid:54)= ∅ be a non-empty set, then a subset of the form R ⊆ X×X
said to be a relation on X. And in this case it is customary to write
xRy instead of (x,y) ∈ R. This notation will be used primarily for
partial orders and equivalence relations (see below).
• Consider any non-empty set X (cid:54)= ∅ again. Then a relation ∼ on X is
saidtobeanequivalence relationonX, iffitisreflexive, symmetric
and transitive. Formally that is for any x, y and z ∈ X we get
x = y =⇒ x ∼ y
x ∼ y =⇒ y ∼ x
x ∼ y, y ∼ z =⇒ x ∼ z
10 0 Prelude