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.)

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.

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.

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And for those who speak Belorussian, I am now pleased to offer:

Click here to read Belorussian translation