**Unformatted text preview: **COMMUTATIVE ALGEBRA
PETE L. CLARK Contents
Introduction
0.1. What is Commutative Algebra?
0.2. Why study Commutative Algebra?
0.3. Acknowledgements
1. Commutative rings
1.1. Fixing terminology
1.2. Adjoining elements
1.3. Ideals and quotient rings
1.4. The monoid of ideals of R
1.5. Pushing and pulling ideals
1.6. Maximal and prime ideals
1.7. Products of rings
1.8. A cheatsheet
2. Galois Connections
2.1. The basic formalism
2.2. Lattice Properties
2.3. Examples of Antitone Galois Connections
2.4. Antitone Galois Connections Decorticated: Relations
2.5. Isotone Galois Connections
2.6. Examples of Isotone Galois Connections
3. Modules
3.1. Basic definitions
3.2. Finitely presented modules
3.3. Torsion and torsionfree modules
3.4. Tensor and Hom
3.5. Projective modules
3.6. Injective modules
3.7. Flat modules
3.8. Nakayama’s Lemma
3.9. Ordinal Filtrations and Applications
3.10. Tor and Ext
3.11. More on flat modules
3.12. Faithful flatness
4. First Properties of Ideals in a Commutative Ring
4.1. Introducing maximal and prime ideals
4.2. Radicals
Date: December 14, 2013.
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83 2 PETE L. CLARK 4.3. Comaximal ideals
4.4. Local rings
4.5. The Prime Ideal Principle of Lam and Reyes
4.6. Minimal Primes
5. Examples of Rings
5.1. Rings of numbers
5.2. Rings of continuous functions
5.3. Rings of holomorphic functions
5.4. Polynomial rings
5.5. Semigroup algebras
6. Swan’s Theorem
6.1. Introduction to (topological) vector bundles
6.2. Swan’s Theorem
6.3. Proof of Swan’s Theorem
6.4. Applications of Swan’s Theorem
6.5. Stably Free Modules
6.6. The Theorem of Bkouche and Finney-Rotman
7. Localization
7.1. Definition and first properties
7.2. Pushing and pulling via a localization map
7.3. The fibers of a morphism
7.4. Commutativity of localization and passage to a quotient
7.5. Localization at a prime ideal
7.6. Localization of modules
7.7. Local properties
7.8. Local characterization of finitely generated projective modules
8. Noetherian rings
8.1. Chain conditions on partially ordered sets
8.2. Chain conditions on modules
8.3. Semisimple modules and rings
8.4. Normal Series
8.5. The Krull-Schmidt Theorem
8.6. Some important terminology
8.7. Introducing Noetherian rings
8.8. Theorems of Eakin-Nagata, Formanek and Jothilingam
8.9. The Bass-Papp Theorem
8.10. Artinian rings: structure theory
8.11. The Hilbert Basis Theorem
8.12. The Krull Intersection Theorem
8.13. Krull’s Principal Ideal Theorem
8.14. The Dimension Theorem, following [BMRH]
8.15. The Artin-Tate Lemma
9. Boolean rings
9.1. First Properties
9.2. Boolean Algebras
9.3. Ideal Theory in Boolean Rings
9.4. The Stone Representation Theorem
9.5. Boolean Spaces 86
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172 COMMUTATIVE ALGEBRA 9.6. Stone Duality
9.7. Topology of Boolean Rings
10. Associated Primes and Primary Decomposition
10.1. Associated Primes
10.2. The support of a module
10.3. Primary Ideals
10.4. Primary Decomposition, Lasker and Noether
10.5. Irredundant primary decompositions
10.6. Uniqueness properties of primary decomposition
10.7. Applications in dimension zero
10.8. Applications in dimension one
11. Nullstellens¨
atze
11.1. Zariski’s Lemma
11.2. Hilbert’s Nullstellensatz
11.3. The Real Nullstellensatz
11.4. The Combinatorial Nullstellensatz
11.5. The Finite Field Nullstellensatz
11.6. Terjanian’s Homogeneous p-Nullstellensatz
12. Goldman domains and Hilbert-Jacobson rings
12.1. Goldman domains
12.2. Hilbert rings
12.3. Jacobson Rings
12.4. Hilbert-Jacobson Rings
13. Spec R as a topological space
13.1. The Zariski spectrum
13.2. Properties of the spectrum: quasi-compactness
13.3. Properties of the spectrum: separation and specialization
13.4. Irreducible spaces
13.5. Noetherian spaces
13.6. Hochster’s Theorem
13.7. Rank functions revisited
14. Integrality in Ring Extensions
14.1. First properties of integral extensions
14.2. Integral closure of domains
14.3. Spectral properties of integral extensions
14.4. Integrally closed domains
14.5. The Noether Normalization Theorem
14.6. Some Classical Invariant Theory
14.7. Galois extensions of integrally closed domains
14.8. Almost Integral Extensions
15. Factorization
15.1. Kaplansky’s Theorem (II)
15.2. Atomic domains, (ACCP)
15.3. EL-domains
15.4. GCD-domains
15.5. GCDs versus LCMs
15.6. Polynomial rings over UFDs
15.7. Application: the Sch¨onemann-Eisenstein Criterion 3 174
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247 4 PETE L. CLARK 15.8. Application: Determination of Spec R[t] for a PID R
15.9. Power series rings over UFDs
15.10. Nagata’s Criterion
16. Principal rings and B´ezout domains
16.1. Principal ideal domains
16.2. Some structure theory of principal rings
16.3. Euclidean functions and Euclidean rings
16.4. B´ezout domains
17. Valuation rings
17.1. Basic theory
17.2. Ordered abelian groups
17.3. Connections with integral closure
17.4. Another proof of Zariski’s Lemma
17.5. Discrete valuation rings
18. Normalization theorems
18.1. The First Normalization Theorem
18.2. The Second Normalization Theorem
18.3. The Krull-Akizuki Theorem
19. The Picard Group and the Divisor Class Group
19.1. Fractional ideals
19.2. The Ideal Closure
19.3. Invertible fractional ideals and the Picard group
19.4. Divisorial ideals and the Divisor Class Group
20. Dedekind domains
20.1. Characterization in terms of invertibility of ideals
20.2. Ideal factorization in Dedekind domains
20.3. Local characterization of Dedekind domains
20.4. Factorization into primes implies Dedekind
20.5. Generation of ideals in Dedekind domains
20.6. Finitely generated modules over a Dedekind domain
20.7. Injective Modules
21. Pr¨
ufer domains
21.1. Characterizations of Pr¨
ufer Domains
21.2. Butts’s Criterion for a Dedekind Domain
21.3. Modules over a Pr¨
ufer domain
22. One Dimensional Noetherian Domains
22.1. Finite Quotient Domains
23. Structure of overrings
23.1. Introducing overrings
23.2. Overrings of Dedekind domains
23.3. Elasticity in Replete Dedekind Domains
23.4. Overrings of Pr¨
ufer Domains
23.5. Kaplansky’s Theorem (III)
23.6. Every commutative group is a class group
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316 COMMUTATIVE ALGEBRA 5 Introduction
0.1. What is Commutative Algebra?
Commutative algebra is the study of commutative rings and attendant structures,
especially ideals and modules.
This is the only possible short answer I can think of, but it is not completely
satisfying. We might as well say that Hamlet, Prince of Denmark is about a fictional royal family in late medieval Denmark and especially about the title (crown)
prince, whose father (i.e., the King) has recently died and whose father’s brother
has married his mother (i.e., the Queen). Informative, but not the whole story! 0.2. Why study Commutative Algebra?
What are the purely mathematical reasons for studying any subject of pure mathematics? I can think of two:
I. Commutative algebra is a necessary and/or useful prerequisite for the study
of other fields of mathematics in which we are interested.
II. We find commutative algebra to be intrinsically interesting and we want to
learn more. Perhaps we even wish to discover new results in this area.
Most beginning students of commutative algebra can relate to the first reason:
they need, or are told they need, to learn some commutative algebra for their study
of other subjects. Indeed, commutative algebra has come to occupy a remarkably
central role in modern pure mathematics, perhaps second only to category theory
in its ubiquitousness, but in a different way. Category theory provides a common
language and builds bridges between different areas of mathematics: it is something
like a circulatory system. Commutative algebra provides core results that other results draw upon in a foundational way: it is something like a skeleton.
The branch of mathematics which most of all draws upon commutative algebra
for its structural integrity is algebraic geometry, the study of geometric properties
of manifolds and singular spaces which arise as the loci of solutions to systems of
polynomial equations. In fact there is a hard lesson here: in the 19th century algebraic geometry split off from complex function theory and differential geometry
as its own discipline and then burgeoned dramatically at the turn of the century
and the years thereafter. But by 1920 or so the practitioners of the subject had
found their way into territory in which “purely geometric” reasoning led to serious errors. In particular they had been making arguments about how algebraic
varieties behave generically, but they lacked the technology to even give a precise
meaning to the term. Thus the subject ultimately proved invertebrate and began
to collapse under its own weight. Starting around 1930 there began a heroic shoring
up process in which the foundations of the subject were recast with commutative
algebraic methods at the core. This was done several times over, in different ways,
by Zariski, Weil, Serre and Grothendieck, among others. For the last 60 years it 6 PETE L. CLARK has been impossible to deeply study algebraic geometry without knowing commutative algebra – a lot of commutative algebra. (More than is contained in these notes!)
The other branch of mathematics which draws upon commutative algebra in an
absolutely essential way is algebraic number theory. One sees this from the beginning in that the Fundamental Theorem of Arithmetic is the assertion that the ring
Z is a unique factorization domain (UFD), a basic commutative algebraic concept.
Moreover number theory was one of the historical sources of the subject. Notably
the concept of Dedekind domain came from Richard Dedekind’s number-theoretic
investigations. Knowledge of commutative algebra is not as indispensable for number theory (at least, not at the beginning) as it is for algebraic geometry, but such
knowledge brings a great clarifying effect to the subject.
In fact the interplay among number theory, algebraic geometry and commutative
algebra flows in all directions. What Grothendieck did in the 1960s (with important contributions from Chevalley, Serre and others) was to create a single field of
mathematics that encompassed commutative algebra, classical algebraic geometry
and algebraic number theory: the theory of schemes. As a result, most contemporary number theorists are also partly commutative algebraists and partly algebraic
geometers: we call this cosmopolitan take on the subject arithmetic geometry.
There are other areas of mathematics that draw upon commutative algebra in
important ways. To mention some which will show up in later in these notes:
•
•
•
• Differential topology.
General topology.
Invariant theory.
Order theory. The task of providing a commutative algebraic foundation for algebraic geometry
– or even the single, seminal text of R. Hartshorne – is a daunting one. Happily,
this task has been completed by David Eisenbud (a leading contemporary expert
on the interface of commutative algebra and algebraic geometry) in his text [Eis].
This work is highly recommended. It is also 797 pages long, so contains enough
material for 3 − 5 courses in the subject. It would be folly to try to improve upon,
or even successfully imitate, Eisenbud’s work here, and I certainly have not tried.
I myself am an arithmetic geometer (which, as I tried to explain above, is a sort of
uppity kind of number theorist), so it is not surprising that these notes are skewed
more towards number theory than most introductory texts on commutative algebra.
However for the most part a respectful distance is maintained: we rarely discuss
number theory per se but rather classes of rings that a number theorist would like:
Dedekind domains, valuation rings, B´ezout domains, and so forth.
Just much as I have included some material of interest to number theorists I have
included material making connections to other branches of mathematics, especially
connections which are less traditionally made in commutative algebra texts. In
fact at several points I have digressed to discuss topics and theorems which make COMMUTATIVE ALGEBRA 7 connections to other areas of mathematics:
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• §2 on Galois connections.
§5.2 on rings of continuous functions.
§6 on vector bundles and Swan’s Theorem.
§9 on Boolean rings, Boolean spaces and Stone Duality.
§13 on the topology of prime spectra, including Hochster’s Theorem.
§14.6 on invariant theory, including the Shephard-Todd-Chevalley Theorem. But I do find commutative algebra to be of interest unto itself, and I have tried to
craft a sustained narrative rather than just a collection of results.
0.3. Acknowledgements.
Thanks to Max Bender, Martin Brandenburg, John Doyle, Georges Elencwajg,
Emil Jerabek, Keenan Kidwell, David Krumm, Allan Lacy, Casey LaRue, Stacy
´
Musgrave, Alon Regev, Jacob Schlather, Jack Schmidt, Mariano Su´arez-Alvarez
and Matth´e van der Lee for catching errors1 and making other useful suggestions.
Thanks to Hans Parshall for introducing me to the Stone-Tukey Theorem.
1. Commutative rings
1.1. Fixing terminology.
We are interested in studying properties of commutative rings with unity.
By a general algebra R, we mean a triple (R, +, ·) where R is a set endowed
with a binary operation + : R × R → R – called addition – and a binary operation
· : R × R → R – called multiplication – satisfying the following:
(CG) (R, +) is a commutative group,
(D) For all a, b, c ∈ R, (a + b) · c = a · c + b · c, a · (b + c) = a · b + a · c.
For at least fifty years, there has been agreement that in order for an algebra
to be a ring, it must satisfy the additional axiom of associativity of multiplication:
(AM) For all a, b, c ∈ R, a · (b · c) = (a · b) · c.
A general algebra which satisfies (AM) will be called simply an algebra. A similar
convention that is prevalent in the literature is the use of the term nonassociative
algebra to mean what we have called a general algebra: i.e., a not necessarily
associative algebra.
A ring R is said to be with unity if there exists a multiplicative identity, i.e.,
an element e of R such that for all a ∈ R we have e · a = a · e = a. If e and e0
are two such elements, then e = e · e0 = e0 . In other words, if a unity exists, it is
unique, and we will denote it by 1.
1Of which many, many remain: your name could go here! 8 PETE L. CLARK A ring R is commutative if for all x, y ∈ R, x · y = y · x.
In these notes we will be (almost) always working in the category of commutative rings with unity. In a sense which will shortly be made precise, this means
that the identity 1 is regarded as part of the structure of a ring and must therefore
be preserved by all homomorphisms.
Probably it would be more natural to study the class of possibly non-commutative
rings with unity, since, as we will see, many of the fundamental constructions of
rings give rise, in general, to non-commutative rings. But if the restriction to
commutative rings (with unity!) is an artifice, it is a very useful one, since two
of the most fundamental notions in the theory, that of ideal and module, become
significantly different and more complicated in the non-commutative case. It is
nevertheless true that many individual results have simple analogues in the noncommutative case. But it does not seem necessary to carry along the extra generality of non-commutative rings; rather, when one is interested in the non-commutative
case, one can simply remark “Proposition X.Y holds for (left) R-modules over a
noncommutative ring R.”
Notation: Generally we shall abbreviate x · y to xy. Moreover, we usually do
not use different symbols to denote the operations of addition and multiplication
in different rings: it will be seen that this leads to simplicity rather than confusion.
Group of units: Let R be a ring with unity. An element x ∈ R is said to be a
unit if there exists an element y such that xy = yx = 1.
Exercise 1.1:
a) Show that if x is a unit, the element y with xy = yx = 1 is unique, denoted x−1 .
b) Show that if x is a unit, so is x−1 .
c) Show that, for all x, y ∈ R, xy is a unit ⇐⇒ x and y are both units. d) Deduce
that the units form a commutative group, denoted R× , under multiplication. Example (Zero ring): Our rings come with two distinguished elements, the additive identity 0 and the multiplicative identity 1. Suppose that 0 = 1. Then for
x ∈ R, x = 1 · x = 0 · x, whereas in any rin g 0 · x = (0 + 0) · x = 0 · x + 0 · x, so
0 · x = 0. In other words, if 0 = 1, then this is the only element in the ring. It is
clear that for any one element set R = {0}, 0 + 0 = 0 · 0 = 0 endows R with the
structure of a ring. We call this ring the zero ring.
The zero ring exhibits some strange behavior, such that it must be explicitly excluded in many results. For instance, the zero element is a unit in the zero ring,
which is obviously not the case in any nonzero ring. A nonzero ring in which every
nonzero element is a unit is called a division ring. A commutative division ring
is called a field.
Let R and S be rings (with unity). A homomorphism f : R → S is a map COMMUTATIVE ALGEBRA 9 of sets which satisfies
(HOM1) For all x, y ∈ R, f (x + y) = f (x) + f (y).
(HOM2) For all x, y ∈ R, f (xy) = f (x)f (y).
(HOM3) f (1) = 1.
Note that (HOM1) implies f (0) = f (0 + 0) = f (0) + f (0), so f (0) = 0. Thus
we do not need to explcitly include f (0) = 0 in the definition of a group homomorphism. For the multiplicative identity however, this argument only shows that if
f (1) is a unit, then f (1) = 1. Therefore, if we did not require (HOM3), then for
instance the map f : R → R, f (x) = 0 for all x, would be a homomorphism, and
we do not want this.
Exercise 1.2: Suppose R and S are rings, and let f : R → S be a map satisfying (HOM1) and (HOM2). Show that f is a homomorphism of rings (i.e., satisfies
also f (1) = 1) iff f (1) ∈ S × .
A homomorphism f : R → S is an isomorphism if there exists a homomorphism
g : S → R such that: for all x ∈ R, g(f (x)) = x; and for all y ∈ S, f (g(y)) = y.
Exercise 1.3: Let f : R → S be a homomorphism of rings. Show TFAE:
(i) f is a bijection.
(ii) f is an isomorphism.
Remark: In many algebra texts, an isomorphism of rings (or groups, etc.) is defined to be a bijective homomorphism, but this gives the wrong idea of what an
isomorphism should be in other mathematical contexts (e.g. for topological spaces).
Rather, having defined the notion of a morphism of any kind, one defines isomorphism in the way we have above.
Exercise 1.4: a) Suppose R and S are both rings on a set containing exactly one
element. Show that there is a unique ring isomorphism from R to S. (This is a
triviality, but explains why are we able to speak of the zero ring, rather than
simply the zero ring associated to one element set. We will therefore denote the
zero ring just by 0.)
b) Show that any ring R admits a unique homomorphism to the zero ring. One
says that the zero ring is the final object in the category of rings.
Exercise 1.5: Show that for a not-necessarily-commutative-ring S there exists a
unique homomorphism from the ring Z of integers to S. (Thus Z is the initial object in the category of not-necessarily-commutative-rings. It follows immediately
that it is also the initial object in the category of rings.)
A subring R of a ring S is a subset R of S such that
(SR1) 1 ∈ R.
(SR2) For all r, s ∈ R, r + s ∈ R, r − s ∈ R, and rs ∈ R. 10 PETE L. CLARK Here (SR2) expresses that the subset R is an algebra under the operations of addition and multiplication defined on S. Working, as we are, with...

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