Rings and Modules MAS427

This is a level 4 course, also available for MSc.
Lectures are on Tuesday at 1 in Mathematics 103 and 2 in Mathematics B17.
The lecturer is Professor Bill Jackson.

Invitation

This course is an introduction to module theory. Module theory was built up during the first half of the twentieth century, to collect together algebraic ideas that were important for various applications in group theory, number theory, geometry and algebraic topology among other places. Unfortunately there won't be time in this course to go into any of these applications in detail. But the theory of modules makes good sense on its own, and it is one of the most elegant parts of modern algebra.

The basic idea is that abelian groups are very well behaved objects, and we can understand other kinds of mathematical object better if we can somehow relate them to abelian groups. One common way of doing this is to let the mathematical object act on suitable abelian groups as a collection of operators, i.e. as endomorphisms of the groups. In this way the object generates a ring R, and we can look at the family of all abelian groups on which this ring R acts; these groups are the R-modules. A huge amount of information about R can be extracted not just from the modules but from the homomorphisms between them; so we shall have a lot of diagrams with arrows. One of the most unexpected features of this theory (some people originally complained that it was 'theology not mathematics') was the powerful and simple way that it uses finiteness assumptions (which say that some set is finite without saying exactly how large).

All courses in this subject ultimately go back to the lecture notes of the great German algebraist Emmy Noether at the University of Goettingen in 1927/8. Our version of the course is based on notes of Wilfred Hodges.

Emmy Noether, 1882-1935

Syllabus

Introduction to module theory, starting from the definition of module: free, flat, projective and injective modules, products, coproducts, tensor products, exactness and the Hom functor will be covered. The notion of a ring will be assumed.
Structure theorems: chain conditions on rings and modules, Noetherian rings, Artinian rings, Artin-Wedderburn Theorem and the structure of finitely generated modules over principal ideal domains.

Lectures and coursework

A copy of Wilfred Hodge's notes for this course, is available as a pdf file. This year's courseworks will be online here as soon as they become available.

Exercises 1 (pdf file)

Exercises 2 (pdf file)

Exercises 3 (pdf file)

Exercises 4 (pdf file)

Exercises 5 (pdf file)

Key objectives

If you have a secure grasp of the key objectives, you should have no trouble passing the exam. But the exam will certainly contain things not in the key objectives.

You should know the following definitions:

abelian group, ring, ideal, module,
cartesian product of sets, direct product and direct sum of modules,
Hom_R(M,N),
free, projective and injective module,
tensor product of modules (by universal mapping property), including proof of uniqueness,
flat module,
artinian and noetherian module (both in terms of chain conditions and in terms of minimal or maximal submodules),
algebra over a commutative ring,
completely reducible module, semisimple ring.

You should be able to state:

Zorn's lemma,
characterisations of free and projective module,
explicit construction of tensor product,
construction of algebras from tensor product of algebras,

You should be able to prove:

correspondence theorem and isomorphism theorem for modules,
behaviour of free, projective and injective modules with respect to direct sums,
behaviour of tensor product with respect to module homomorphisms and direct sums,
associative and commutative laws for tensor product,
behaviour of flat modules with respect to direct sums,
all projective modules are flat,
a module is noetherian if and only if all submodules are finitely generated,
behaviour of artinian and noetherian modules with respect to submodules and quotients,
if a ring is left or right artinian or noetherian, then so are its finitely generated left or right modules,
equivalence of the characterisations of completely reducible module,
equivalence of the characterisations of a semisimple ring,
the Artin-Wedderburn theorem.

Assessment

Assessment is entirely by examination in the summer. The examination will last THREE HOURS and will contain SIX questions. The rubric will state:

You may attempt as many questions as you wish and all questions carry equal marks. Except for the award of a bare pass, only the best FOUR questions answered will be counted. Calculators may not be used in this examination.

Five or six lots of coursework will be handed out during the course. Submitted coursework will be marked but will not count in the assessment.

Author: Bill Jackson
Last updated 13 January 2004