C2.5 Non-Commutative Rings (2022-23)

General Prerequisites:
All the material of A3 Rings and Modules is essential: Basic properties of rings and modules. Ideals, prime ideals. Principal ideal rings, unique factorization rings, Euclidean rings. Finite fields. Modules over Euclidean rings.

Recommended material:
From B2.1 Introduction to Representation Theory: semisimple modules and algebras, the Artin - Wedderburn theorem.
From B2.2 Commutative Algebra: Noetherian rings and modules. Hilbert's basis theorem. Krull dimension.
Course Term: Hilary
Course Lecture Information: 16 lectures.
Course Weight: 1
Course Level: M
Assessment Type: Written Examination
Course Overview:
This course builds on Algebra 2 from the second year. We will look at several classes of non-commutative rings and try to explain the idea that they should be thought of as functions on "non-commutative spaces". Along the way, we will prove several beautiful structure theorems for Noetherian rings and their modules.
Learning Outcomes:
Students will be able to appreciate powerful structure theorems, and be familiar with examples of non-commutative rings arising from various parts of mathematics.
Course Synopsis:
Examples of non-commutative Noetherian rings: enveloping algebras, rings of differential operators, group rings of polycyclic groups. Filtered and graded rings. (3 hours)

Jacobson radical in general rings. Jacobson's density theorem. Artin-Wedderburn. (3 hours)

Ore localisation. Goldie's Theorem on Noetherian domains. (3 hours)

Minimal prime ideals and dimension functions. Rees rings and good filtrations. (3 hours)

Bernstein's Inequality and Gabber's Theorem on the integrability of the characteristic variety. (4 hours)