1.  Introduction:  This course provides an introduction to a flourishing and fundamental area of contemporary mathematics. In the course the basic results of non-commutative ring theory will be proved and the following topics are included: simple and semi-simple rings, division rings, radicals and the elementary theory of modules. Algebra I and II are prerequisites.

2.  Timetable Hours, Credits, Assessments:  The course is a 20 UCU unit of 33 lectures, 5 problem sheets and occasional problem sessions. Assessment is by marked homework (20%) and examination (80%).

3.  Overview:  Ring theory has many applications in mathematics and thanks to the re-discovery of its connections to physics in the last fifteen years is playing an ever more important role. The aim of the course is to introduce the main ideas and notions of the topic and to explain some of the machinery and basic results. The course demonstrates the power of the application of linear algebra which students have become familiar with in previous years.

The course starts from introducing the main notions and showing a variety of examples. The idea of a module and its endomorphism ring is crucial for the theory. It then follows an exposition of the classical results of the theory such as Schur's lemma and Maschke's theorem. Their role in the topic can not be overestimated. One of the most powerful and efficient methods in non-commutative ring theory is the analysis of their representations. This occupies a considerable part of the course. Another important idea which will be investigated is the radical of a ring.

The course ends with variety of examples and applications. 5 Exercise sheets help students to understand the notions, machinery and particular problems of the course.

4.  Recommended Reading:  All of the following should be in the Library and many other books cover the material of the course.

1. T Y Lam, A First course in Non-commutative Rings, Springer-Verlag, Berlin, 1991.

2. I Herstein, Noncommutative Rings.

3. I Adamson, Rings, Modules and Algebras, Oliver & Boyd, Edinburgh, 1971.

4. N H Mccoy, The Theory of Rings, Macmillan, N.Y.-London, 1964.

Rings, modules, algebras. (2 lectures)

Opposite rings. (1 lecture)

Group rings. (1 lecture)

Division rings. (1 lecture)

Endomorphism ring of abelian groups. (1 lecture)

Submodules. Module homomorphisms. (1 lecture)

Jordan-Hölder theorem. (1 lecture)

Irreducible modules. Schur's lemma. (1 lecture)

Free modules. Endomorphism rings of free modules. (2 lectures)

Simple rings with a minimal ideal. (1 lecture)

Wedderburn-Artin theorem. (2 lectures)

Automorphisms of simple rings. (1 lecture)

Structure of division rings. (2 lectures)

Completely reducible modules. (1 lecture)

Modules over simple artinian rings. (1 lecture)

Semisimple rings. (2 lectures)

Mascke's theorem. (1 lecture)

Jacobson radical. (2 lectures)

Structure of artinian rings. (2 lectures)

Structure theorem for simple finite dimensional algebras. (2 lectures)

Group representation theory via group rings. (3 lectures)

Applications to matrix groups and group representations. Examples. (2 lectures)