Cornell Math - MATH 632, Spring 1999

MATH 632 — Spring 1999
Noncummutative Algebra

Instructor: Arkady Berenstein

Time:  MWF 11:15-12:05

Room: UH 204

The prerequisite for this course is a knowledge of commutative and abstract algebra at the level of MATH 631.

In modern algebra ring theory is of great importance. Historically, the rings emerged from various branches of mathematics: number theory (rings of integers, orders, Hecke rings), representation theory (group rings, enveloping algebras, rings of matrices), differential geometry and functional analysis (algebras of functions and operators, e.g. differential operators), homological algebra (cohomology rings, abelian categories, Grothendieck K-groups).

One of the main goals of the course is to give a systematic introduction to noncommutative ring theory and to emphasize the role of this theory in algebra (and, if time permits, in geometry).

I will mostly follow the textbook by Farb and Dennis, Noncommutative Algebra. More precisely, I will cover the first four chapters of the text thoroughly: semisimple modules and rings, the Wedderburn structure theorem, the Jacobson radical, central simple algebras, and the Brauer group. I then plan to emphasize the following additional topics as time permits:

• Relationship with category theory, especially with abelian categories;
• Group cohomology and the cohomological interpretation of the Brauer group;
• Representation theory of finite groups.

Other reference books:

S. Lang, Algebra, Addison-Wesley, 1993.

T. Lam, A first course in noncommutative rings, Springer-Verlag, 1991.

K. Goodearl, R. Warfield, Jr., An introduction to noncommutative noetherian rings, New York, 1989.