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MC445 Ring Theory

MC 445 Ring Theory

Credits: 20

Convenor: Dr. J.F. Watters

Semester: 2

Prerequisites:	essential: MC242 or MC341; MC249 or MC344=MC382
Assessment:	Continual assessment: 10%	Three hour exam in May/June: 90%

Lectures:	36	Classes:	10
Tutorials:	none	Private Study:	104
Labs:	none	Seminars:	none
Project:	none	Other:	none
Total:	150

Explanation of Pre-requisites

It is desirable that students have experience of group theory and elementary ring theory before taking this module. Group theory will have given experience of working with cosets and factor groups and, if the Level 3 module has been studied, knowledge of isomorphism theorems. The rings/abstract algebra modules include basic material on rings and ideals which will be revised at the start of this module.

Course Description

The course explores the theme of structure in non-commutative rings. By structure we mean being able to build rings from what we regard as concrete examples using some tangible process. The process we use here is the algebraic version of the Cartesian product, called the direct sum, and the concrete examples are matrix rings. The entries in these matrix rings have to be allowed to come not merely from fields, but the non-commutative analogue, division rings. In this course we explore the conditions we need to impose on a ring in order that it be expressible as a direct sum of a finite number of matrix rings over division rings. The conditions needed embody finiteness in a certain way and also require the ring to be free of ideals of a particular type. The division rings arise by considering simple modules; the awkward ideals which we wish to be free of determine the Jacobson radical; and the finiteness condition is imposed on descending chains of ideals.

Aims

To develop the basic theory of modules over rings leading to an understanding of the structure of certain classes of non-commutative rings.

Objectives

To be familiar with the basic algebra of modules and rings.
To understand the direct sum construction.
To connect simple modules and division rings through Schur's Lemma.
To know the elementary properties of Artinian modules.
To define the prime and Jacobson radicals in rings.
To establish properties of these two radicals.
To prove the Wedderburn-Artin Theorem.

Transferable Skills

Developing understanding of abstraction and algebraic structure.
Ability to connect module concepts with ring concepts.
Understanding of proofs.
Ability to apply theorems to solve problems.
Written presentation of algebraic arguments.

Syllabus

Rings, ideals and homomorphisms.
Modules; submodules, factor modules and homomorphisms; cyclic, simple and semisimple modules; (if time; free, projective, and injective module).
Finiteness conditions on rings and modules.
Simple, prime, and primitive rings.
The prime radical.
The Jacobson radical.
Structure of semi-simple rings.

Reading list

Background:

Details of Assessment

Coursework will be set weekly and marks gained will provide the coursework component for the assessment $(10\%)$ . The midsummer examination will be of three hours duration. It will contain 6 questions with full marks attainable for complete answers to 4 questions. The examination mark gives $90\%$ of the assessment for the module.

Next: MC480 Mathematics Project Up: Year 4 Previous: MC442 Differential Geometry

S. J. Ambler
11/20/1999