Ring Theory

These notes are designed for students in the honours year who have a good grounding in both group theory and linear algebra. We begin the ring theory with some general foundations and then the theory of Euclidean rings.

An algebra is a ring that also has the structure of a vector space over some field (most rings of any importance are algebras too) and the advantage of studying algebras is that we can make use of the vector space structure.

From the third chapter the focus is on non-commutative algebras and their modules. The goal is to prove the Wedderburn Structure Theorem for "semi-simple algebras with descending chain condition on right ideals". These algebras include group algebras of finite groups over C and the Wedderburn theorem enables us to prove the Fundamental Theorem of Characters.

The aim of these notes is not to get to the Wedderburn theorem as quickly as possible, but rather to take a more leisurely and circuitous tour of radicals in rings, following to some extent the route laid down by Divinsky in his book Rings and Radicals.

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