# MA6H8 Ring Theory

**Lecturer:** Marco Schlichting

**Term(s):** Term 1

**Commitment:** 30 Lectures

**Assessment:** Oral Exam

**Prerequisites:** Familiarity with basic concepts in rings and modules. e.g. from the MA3G6 Commutative algebra course.

**Content**:

- Review of MA377 Rings and Modules
- Jacobson Radical, Artin-Wedderburn, Hopkins theorem
- Central simple Algebras and the Brauer group
- Computation of the Brauer group of the rational numbers
- K_2 and the Brauer group of a field

**Aims**: The broad goal is to classify (non-commutative) Artinian rings, for instance, finite dimensional algebras over a field. We will show that every Artinian ring modulo a certain nilpotent ideal, called the Jacobson radical, is semi-simple. From MA377 Rings and Modules we know that semi-simple rings are finite products of matrix rings over division rings. Every division ring is an algebra over its centre which is a field. Under a suitable operation, the "set" of finite dimensional division algebras with centre F forms a group, called the Brauer group Br(F) of F. We will compute the Brauer group of finite fields, the real and complex numbers, p-adic numbers, and most notably of the rational numbers thereby providing a complete classification of division algebras over these fields.

**Books:**

Benson Farb, R. Keith Dennis: *Noncommutative Algebra* (Graduate Texts in Mathematics), ISBN: 038794057X

Richard Pierce: *Associative Algebras*. Graduate Texts in Mathematics, 88. Springer-Verlag, New York-Berlin, ISBN: 0-387-90693-2

Philippe Gille, Tamas Szamuely: *Central Simple Algebras and Galois Cohomology*. Cambridge University Press, Cambridge*,* ISBN: 978-1-316-60988-0