MATH 6330 - Noncommutative Algebra

Yuri Berest, spring 2014.

The course is intended to be an introduction to methods of noncommutative algebra. Topics will include semisimple rings and modules (Artin-Wedderburn Theorem), module categories and Morita theory, Noetherian rings (quotient rings and Goldie Theorem), basic dimension theory (Krull dimension, global dimension and Gelfand-Kirillov dimension), projective modules (an introduction to algebraic K-theory: $K_0$ and $K_1$ of rings). Examples: polynomial identity rings (Azumaya algebras), enveloping algebras of Lie algebras, rings of differential operators on algebraic varieties. Our guiding principle will be to focus on general ring-theoretic techniques that play a role in other areas of mathematics, mostly in algebraic geometry, algebraic topology and representation theory.

Some basic references:

1. J. C. McConnell and J.C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics 30, AMS, Providence, RI, 2001.
2. Bo Stenstrom, Rings of Quotients. An Introduction to Methods of Ring Theory, Springer-Verlag, New York-Heidelberg, 1975.
3. F. Anderson and K. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics 13, Springer-Verlag, New York, 1992.
4. K. R. Goodearl and R. B. Warfield, An introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts 61, Cambridge, 2004.
5. J. Milnor, Introduction to Algebraic K-theory, Annals of Mathematics Studies 72, Princeton University Press, 1971.