Description
The aim of this project is to study some topics in algebra
which pick up where Algebra II left off. The project will be
oriented towards examples and exercises which you can
partially select yourself and which will form an important
part of your project report later.
In Michaelmas we will cover the following:
- The concept of module over a ring. These are
basically like vector spaces, but over rings instead of
over fields. Unlike vector spaces, they may not always
have a basis. Modules shed a lot of light on group
theory and ring theory. For instance, an abelian group
is just a module over $\mathbb{Z}$, and an ideal in a
ring is just a submodule of the ring over itself.
(Includes: left/right modules over non-commutative
rings, submodules, quotients, bases/free modules, tensor
products).
- Vector spaces. We use the tool of modules
over rings to shed new light on concepts of linear
algebra. For instance, we can really explain
where the determinant or transpose of a matrix come
from.
- Modules over a PID. We will prove a complete
structure theorem for finitely generated modules over a
Principal Ideal Domain. In particular, this classifies
all finite abelian groups, but it also gives a beautiful
proof of Jordan Canonical Form.
Topics for further individual study in the latter stages
of the project may include:
- More on modules over non-commutative rings. Left/right
Noetherian/Artinian modules.
- projective, injective and flat modules. Exact
sequences, rudiments of homological algebra,
- the module theoretic approach to representations of
finite groups.
Prerequisites
Algebra II.
Resources
We will follow Part III (Chapters 10-12) of D. S. Dummit and
R. M. Foote, Abstract algebra (3rd edition or
later). You are encouraged to find other additional or
alternative resources on your own.
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