A lot of the files listed below are in PDF (Adobe Acrobat) format. Alternate versions are in DVI format (produced by TeX; see see here for a DVI viewer provided by John P. Costella) and postscript format (viewable with ghostscript.) Some systems may have some problem with certain of the documents in dvi format, because they use a few German letters from a font that may not be available on some systems. (Three alternate sites for DVI viewers, via FTP, are CTAN, Duke, and Dante, in Germany.)
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:
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.)
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.
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.
(For more material on commutative ring theory,
see also
Chapter Zero of my book
on torsion free modules over Dedekind domains.)
(
Click here for dvi version.)
Projective and injective modules.
Tensor products. Flat modules.
Basic field theory.
Homework Set 1 .
(Click here for DVI format)
(Click here for Postscript)
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.
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.)
Homework Set 2
(Click here for DVI format.)
(Click here for Postscript.)
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.)
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.)
Homework Set 3
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(Click here for Postscript.)
The concept of the product of two rings.
Part of the proof of Fitting's Lemma.
Homework Set 4
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(Click here for Postscript.)
Some examples taken from abelian group theory.
Homework Set 5
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(Click here for Postscript.)
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
(Click here for DVI format.)
(Click here for Postscript.)
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
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(Click here for Postscript.)
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
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(Click here for Postscript.)
In this set, I gave them an interesting Lemma which is part of the theory of semi-simple rings.
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.
Homework Set 9
(Click here for DVI format.)
(Click here for Postscript.)
Fully invariant direct summands.
Prove that a minimal left ideal whose square is not zero is generated by an idempotent.
Homework Set 10 .
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(Click here for Postscript.)
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
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(Click here for Postscript.)
Localization-globalization theorems for modules over commutative rings.
Homework Set 12
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(Click here for Postscript.)
Associated primes and localization.
Homework Set 13
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(Click here for Postscript.)
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
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(Click here for Postscript.)
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
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(Click here for Postscript.)
The categorical definitions of monomorphisms and epimorphisms.
Principles of diagram chasing. (Once again, excuse the missing diagonal arrows.)
The Snake Lemma.
Homework Set 16
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(Click here for Postscript.)
The classification of indecomposable projective modules over an artinian ring.
Natural transformations.
Yoneda's Lemma.
Homework Set 17
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(Click here for Postscript.)
Indecomposable injective modules.
Prove that the endomorphism ring of an indecomposable injective module is a local ring.
Homework Set 18
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(Click here for Postscript.)
Construction of injective modules.
The symmetric algebra determined by a module over a commutative ring.
Homework Set 19
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Problems on tensor products.
Homework Set 20
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(Click here for Postscript.)
The tensor product of two rings.
Homework Set 21
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(Click here for Postscript.)
Flat modules and pure submodules.
The augmentation ideal in a group algebra.
Homework Set 22
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(Click here for Postscript.)
Separable and inseparable field extensions.