Information for Math 557: Ring Theory
Fall 2012

• To improve your ability to write proofs.
• To improve your understanding of and ability to use common modes of algebraic reasoning.
• To learn basic ideas in the theory of commutative noetherian rings and their modules.

• Ideals, varieties, and algorithms, by David Cox, John Little, and Donal O'Shea
• Undergraduate commutative algebra, by Miles Reid
• Introduction to commutative algebra, by Michael Atiyah and Ian Macdonald
• Commutative ring theory, by H. Matsumura
• Commutative Algebra, with a view towards algebraic geometry, by David Eisenbud
• Cohen--Macaulay rings, by Winfried Bruns and Jurgen Herzog

High school algebra, and mathematical maturity in algebraic ideas expected from an undergraduate abstract algebra course including proofs. As a benchmark, you should have seen and thoroughly understood a proof of the First Isomorphism Theorem in some context. This does not mean you have to remember it. (This theorem says that, if h: A to B is a homomorphism, then A/kernel(h) is isomorphic to the image(h).)

I will start from scratch in that no prior knowledge of rings will be assumed, but the pace will be more rapid than in an undergraduate abstract algebra course.

In the first 10--12 weeks, I plan to cover the basic ideas of commutative rings and their modules. This includes prime and maximal ideals, primary decomposition, rings of fractions, modules, chain conditions, integral dependence, and dimension theory.

If there is time, I will cover the theory of modules over a principal ideal domain, canonical forms for matrices from the commutative algebra point of view, and the theory of depth and Cohen--Macaulay rings.

I will closely follow Sharp's book. The optional topics are in Chapters 10, 11, 16, and 17, and the topics I definitely plan to cover are the rest of the book.

In the last 3--5 weeks, I will cover methods to more or less practically compute what we have studied in examples of moderate complexity. (This means too complicated to eyeball but not so complicated that the computer will take years.) I will also cover some of the theory behind these methods.

I will loosely follow the book by Cox, Little, and O'Shea.

Unfortunately, I will not have time to cover any of the theory of noncommutative rings (much of which is useful in the representation theory of groups). In part, this is because no one in the department does research (at least not directly) on noncommutative rings, whereas both algebraic geometry and algebraic number theory rely heavily on commutative algebra.

Also unfortunately, one semester is not enough time to cover everything a researcher in commutative algebra (or number theory or algebraic geometry) should know. Hopefully, this course will give you enough of the basics both to get started in research and to give you a solid foundation for further study.

This is a graduate class. The real tests for whether you learn this material are the Masters Comprehensive Exam or the PhD Qualifying Exam. Grades will reflect whether I think you are prepared for the portions of these exams that will cover ring theory. An A means you are fully prepared, a B that you will be prepared with some additional review and study, and a C that you need significant additional work. I hope everyone will earn an A, but this will require hard work on your part. I think everyone will earn at least a B.

I expect to count the homework as a little over half, the midterm as a little under a sixth, and the final as a little under a third of your grade. But I will take a holistic view and grade based on whether you have demonstrated understanding of commutative algebra and ability to write proofs of commutative algebra facts.