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It would take at least three semesters to teach the minimum amount of algebra that a graduate student ought to know, and probably even four semesters would not be overly long. So for a two semester course, a lot of hard choices have to be made, and these choices give rise to spirited disagreements among algebraists. Here at UH, it's also important to realize that the basic graduate algebra course is probably the most advanced algebra course that a student will ever take. At best, a student who decides to major in algebra may eventually also take a one semester course in ring theory or group theory.
One important consideration for me is that the algebra course should cover all the topics in algebra commonly used by analysts and topologists. This means that it's important to cover topics such as commutative diagrams, the tensor product, functors, and Nakayama's Lemma.
I assume from day one that a student knows about groups, rings, and vector spaces. In fact, I begin the course by defining the concept of a module over a ring, and exploring the ways in which modules over rings without division are more complicated than vector spaces.
It's my belief that just as important as teaching the basic topics is teaching the way algebraists think. From the very beginning I emphasize the question of what sorts of things we look for when we study algebra, with an emphasize on structure and classification theorems.
I want every concept in the course to be motivated. I never want to present any concept as a mere definition accompanied by a handful of trivial theorems. Either I want to be able to give at least a few fairly deep theorems about the topic, or, as in the case of a tool such as commutative diagrams, I want to show a diversity of applications for that tool. Also, I never want students to see any concept as something that is completely abstract. So for every concept, I provide concrete non-trivial examples.
Thus, when I define projective and injective modules, I give some major results on their structure, showing how to classify projective modules over an artinian ring and injective modules over a commutative noetherian ring. (A lot of this is done by students in exercises.)
I think it's important for students to realize that we don't (or shouldn't) prove theorems just for the sake of proving theorems. We are trying to understand what the objects which algebra talks about really look like, to see them in another way.
I consider it very important to include some theorems of real substance in the course. Theorems which state remarkable things about fairly basic concepts, and theorems whose proof involves the use of a number of quite diverse results. Some examples are the theorem that every artinian ring (with identity) is noetherian, the theorem that a commutative noetherian ring is a unique factorization domain if and only if its height-one primes are principal, and the theorem that a module over a commutative noetherian ring has finite length if and only if it is finitely generated and all its associated primes are maximal.
Unfortunately, I am simply unable to prove one of the best theorems of this nature: namely, that representations in characteristic zero of a finite group are completely determined by their characters, and that this leads to a complete classification of all the representations. However, last time I taught the course I spent a period giving an expository lecture on this result, and my student (singular) seemed to follow with interest.
In the very beginning of the course, I talk about finiteness conditions. I present the concept of a module over a ring as a generalization of the concept of a vector space. I talk about what a strong condition finite-dimensionality is for vector spaces. The primary importance of finite dimensionality for a vector space is that it guarantees that every monic endomorphism is surjective and conversely. I illustrate this with the theorem showing that in a finite-dimensional algebras, left zero-divisors, right zero-divisors, left invertible elements, and right-invertible elements all coincide, and then give examples to show that these concepts do not all coincide for rings in general.
Then I explore the question of how one should generalize the concept of finite dimensionality for modules over an arbitrary ring. As the most obvious step, this leads to the concept of finite length for modules and the Jordan-Hölder Theorem. But then I point out that finite length is often a very stringent condition for modules, and so I look for weaker conditions that would give at least part of what finite dimensionality does. Namely, instead of hoping for a condition that would guarantee that monic endomorphisms and surjective endomorphisms are the same, what if we only require that monic endomorphisms be surjective? Or what if we only require the converse? This leads to the concepts of the ascending and descending chain conditions.
From here, it is only natural to go into the theory of semi-simple rings and then the more general theory of rings with minimum condition. In any case, I think that this is really important because I think that the Wedderburn Theorem is the quintessential theorem in algebra, in that it sets the ideal, the goal -- one might almost say that Holy Grail -- that we strive towards in all other parts of algebra, viz. classification theorems. We see this ideal actually achieved in only a few other places in algebra: the Fundamental Theorem of Abelian Groups, the classification of all finite simple groups, the structure of injective modules over a commutative notherian ring. To some extent, Galois Theory is another example, but it never succeeds in answering the ultimate question: to classify all the finite dimensional extensions of a given field. For that, we need Class Field Theory, and even there the answer is incomplete.
A covert purpose is to get students to think from the beginning to think about algebra in concrete terms, to think in terms of specific examples.
And for those who speak Belorussian, I am now pleased to offer:
Click here to read Belorussian translation
(has been provided by Webhostingrating)
I'm as much an admirer of slick proofs as the next guy, and I'm pretty
proud of some of the slick proofs I've promolgated. I think it's
important for a graduate student to see many slick proofs, just as it's
important for someone learning to play jazz trumpet to listen to many
recordings of Louis Armstrong and Miles Davis.
But a slick proof usually
doesn't reveal the real reason why a theorem is true in the way that a
clumsy proof can. Furthermore, a slick proof is in some sense a lie, in
that it conceals the lengthy, stumbling process by which one
actually finds the proof of a theorem.
In this proof of the
Jordan-Hölder Theorem, I attempt to simulate the actual process of
discovery by going through a bunch of special cases, which gradually
become more and more general. The first time I taught this, I presented
the installments of the proof in three successive days of class.
It is shown that a surjective endomorphism of a noetherian module
must be an automorphism.
Then Fitting's Lemma is proved,
and from this it is proved that the endomorphism ring
of an indecomposable module with finite length
is a local ring.
A not necessarily commutative ring
is a local ring if and only if the set of non-invertible
elements is closed under addition.
Standard equivalent characterizations of the Jacobson radical.
This covers the basic facts about factorial domains, including
Gauss's Lemma and the fact that a commutative noetherian domain
is factorial if and only if its height-one prime ideals are all
principal.
Neither Lang nor Hungerford seem to have noticed how simple Gauss's
Lemma is. They both give the clumsy traditional proof. But since a
polynomial in R[X] is primitive if and only if it does not
belong to pR[X] for any principal prime ideal p
of R, Gauss's Lemma just comes down to the fact that if
p is a prime ideal in a commutative ring R, then
the ideal pR[X] generated by p in the polynomial
ring is also prime, and this reduces to the trivial assertion that if
D is an integral domain, then D[X] is also an
integral domain.
An ideal in an integral domain is projective
if and only if it is invertible.
The injective envelope E of a module M can be
characterized by the property that every monomorphism
from M into an injective module Q
extends to a monomorphism from E
into Q.
Consider a non-faithful (i.e. "bounded") p-primary module over a principal
ideal domain. (In particular, a finitely generated p-primary module
is not faithful, assuming that p is not (0).) Then any cyclic summand
of maximal "order" (i.e. with minimal non-trivial annihilator) is a
direct summand. This, of course, is a basic step in proving the usual
structure theorem for finitely generated modules over a principal ideal
domain. (A few diagonal lines should be added to the figure here.)
A few standard examples of functors defined on the category of modules
over a given ring.
I never liked proving the Wedderburn-Artin Theorem by means of the
Jacobson Density Theorem, because it seemed unreasonable to have to
prove something as powerful as the density theorem just to use it
in the finite-dimensional case. On the other hand, the proof using
matrices is too computational for my tastes. Finally it occurred to me
that the Wedderburn-Artin Theorem is an example of Morita Equivalence,
and I wondered if it would be in fact possible to derive a simple-minded
version of Morita Equivalence in the basic graduate algebra course.
Finally, I came up with a fairly simple proof of the following theorem:
A left artinian ring (with identity) is also right artinian and
left and right noetherian. Thus it has finite length as a left
or right module over itself.
These are notes I used to teach from, given in semi-disorganized fashion.
The first few are fairly sketchy,
but last four contain pretty much complete statements of theorems,
definitions and such (but very few proofs).
They represent only a portion (but a large one)
of the total course,
namely that portion where I did not follow the book closely
and did not have other notes available.
To get an overview of the whole course,
refer to the syllabus.
This first set of notes is extremely sketchy.
I start the course by giving the basic concepts of module theory
-- direct sums and summands, quotient modules.
I emphasize the comparison to vector spaces.
Principal ideal domains are introduced as examples of rings
over which modules are really nice.
Divisibility and unique factorization domains.
Noetherian and artinian modules
as a generalization of finite-dimensional vector spaces.
The localization of a module with respect to a multiplicative
subset of the center of the ring.
Additive functors.
Associated primes for a module over a commutative ring.
Characterization of modules of finite length
over a commutative ring.
Projective and injective modules.
Tensor products. Flat modules.
Basic field theory.
Homework Set 1 .
This is a pretty demanding first assignment,
not because the proofs are hard, but because it's hard for students
to know exactly what it is they're supposed to prove.
It's pretty much guaranteed to bring students into my office
and also to get rid of students who are not prepared
to do a lot of real thinking about the course.
Students are asked to prove a lot of standard results.
First, that an endomorphism of a vector space V
gives that space the structure of a K[X]-module,
where K is the scalar field.
Homework Set 2
Prove that in a finite-dimensional algebra
the left zero-divisors coincide with the right zero-divisors,
and are also the same as the left and right invertible elements.
(I want students to start thinking about the implications
of finite-dimensionality,
which will later be generalized to the discussion of chain conditions
on modules.)
Homework Set 3
The concept of the product of two rings.
Part of the proof of Fitting's Lemma.
Homework Set 4
Some examples taken from abelian group theory.
Homework Set 5
Here I tried to get students to see what modules look like
over the ring of upper-triangular two-by-two matrices
over a field.
Students couldn't figure this one out at all.
Homework Set 6
The structure of a module over the product of two rings.
Prove that an surjective endomorphism of a module with Maximum Condition
is an automorphism.
Likewise for a monomorphism of a module with Minimum Condition.
(Students don't yet know the words
noetherian and artinian at this point.
I prefer to have the think about the Maximum and Minimum Condition
before I introduce these concepts in terms of chain conditions.)
Homework Set 7
This was another attempt to get students to think about rings
defined by triangular matrices.
They still couldn't figure out how to approach it.
I want to get students to think of a module
as something concrete and describable,
rather than as an amorphous abstraction.
Homework Set 8
In this set, I gave them an interesting Lemma
which is part of the theory of semi-simple rings.
Homework Set 9
Fully invariant direct summands.
Prove that a minimal left ideal whose square is not zero
is generated by an idempotent.
Homework Set 10 .
The characterization of a local ring.
Prove that the group of rational numbers with square free denominators
is generated by the set of fractions 1/p, for p prime.
An example of an indecomposable torsion-free abelian group
of rank 2.
Unfortunately, this group was more than students could deal with.
I want them to have some examples of non-unique direct sum decompositions
before we discuss the Krull-Schmidt Theorem.
Homework Set 11
Localization-globalization theorems
for modules over commutative rings.
Homework Set 12
Associated primes and localization.
Homework Set 13
Push-outs and pullbacks.
(Diagonal arrows are omitted from the diagrams here,
since AmsTex can't draw them.
You have to just imagine them.)
Homework Set 14
The prime ideals in an infinite product of copies of a field.
The ring above is, of course, a classic example of a commutative
von Neumann-regular ring.
The second problem asks students to prove equivalent characterizations
of commutative von Neumann regular rings.
Prove that a height-zero prime
in a commutative not necessarily noetherian ring
consists of zero-divisors.
Homework Set 15
The categorical definitions of monomorphisms and epimorphisms.
Principles of diagram chasing.
(Once again, excuse the missing diagonal arrows.)
The Snake Lemma.
Homework Set 16
The classification of indecomposable projective modules
over an artinian ring.
Natural transformations.
Yoneda's Lemma.
Homework Set 17
Indecomposable injective modules.
Prove that the endomorphism ring of an indecomposable injective module
is a local ring.
Homework Set 18
Construction of injective modules.
The symmetric algebra determined by a module over a commutative ring.
Homework Set 19
Problems on tensor products.
Homework Set 20
The tensor product of two rings.
Homework Set 21
Flat modules and pure submodules.
The augmentation ideal in a group algebra.
Homework Set 22
Separable and inseparable field extensions.
No matter what textbook is used, this syllabus requires a whole lot
of jumping around. I used Hungerford most recently, because in previous
years students had complained vociferously about Lang. But I would never
use Hungerford again. Frankly, I think the book sucks. Hungerford
seems to have
a talent for making all the wrong choices. In particular, the choice of
defining all the concepts in terms of rings without identity is alone
enough to make the book unusable, in my opinion. This mistake
screws up almost every part of the book where rings are relevant.
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Let a ring R be isomorphic as an R-module to a direct sum
of n copies
of a left R-module L. Let D be the ring of R-endomorphisms of L.
Then L is free as a left D-module with rank n. Furthermore, the ring
of D-endomorphisms of L is isomorphic to R. (It follows, of course,
that R is isomorphic to the opposite ring of
the ring of n by n matrices with entries in D.)
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A Reading List
Most of all, my criterion for this list has been
that the books be interesting.
Secondly, I've tried to list books that are good as references
--- the books that I turn to when I'm looking for a standard result.
The Origins of Modern Algebra
This very sketchy history is based completely
on the historical notes in Bourbaki.
Detailed Course Outline
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(For more material on commutative ring theory,
see also
Chapter Zero of my book
on torsion free modules over Dedekind domains.)
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Homework Problems
Many of these problems, especially in the beginning, are extremely hard
for students.
The important thing is not so much that they get them right,
but that they think about them and work on them.
If a student can really think about a problem quite a bit,
then it will be much easier to understand it
when I talk about it in class or during my office hours.
On the other hand, if problems seem impossibly difficult,
then students will decide
that there's no point in even trying to work on them.
Unfortunately, this seemed to be the case
for some of the problems below.
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Next, that the set of endomorphisms of an abelian group
forms a ring.
That a cyclic module over a ring is isomorphic
to the quotient of the ring modulo a left ideal.
That a set S of a module is a maximal linearly independent
subset of a module M if and only if
it generates an essential submodule of M.
(As contrasted with the situation where vector spaces,
where a S is a maximal linearly independent subset
if and only if it spans the whole space).
And, finally, to show that if the rank of a torsion-free module
over a commutative ring is defined as the cardinality
of a maximally linearly independent subset,
then the rank is well defined.
(The proof I had in mind is essentially identical
to the proof of invariance of dimension for vector spaces,
if one bears in mind the previous problem.
These students, though, don't know their linear algebra nearly well enough
to be able to carry such a proof out.)
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Show the correspondence between direct summands of a module
and idempotent endomorphisms.
(The concept of a split exact sequence is introduced here
in disguised form.)
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If a ring R has a minimal left ideal L
which is not contained in any proper (two-sided) ideal,
then R is a finite direct sum of left ideals isomorphic to L
and is a simple ring, i.e. has no proper non-trivial ideals.
Students were also asked to prove the equivalence
of all the different characterizations of the Jacobson radical.
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