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MA4141 Representations of algebras

MA4141 Representations of algebras

Credits: 20 Convenor: Dr. N. J. Snashall Semester: 1
Prerequisites: essential: MA2111(=MC254), MA2161(=MC255)
Assessment: Project and coursework: 100% Examination: 0%
Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Subject Knowledge

Aims

To develop the basic theory of modules over non-commutative rings and give an understanding of the representation theory of an algebra.

Learning Outcomes

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 prove the Wedderburn-Artin Theorem.
To understand the concept of a path algebra.
To know Gabriel's theorem on finite representation type.
To be aware of the connections between ring theoretic and module theoretic results.

Subject Skills

Aims

The ability to present arguments and solutions in a coherent and logical form.
Developing understanding of abstraction and algebraic structure.
The ability to connect module concepts with ring concepts.
Understanding of proofs.
The ability to apply theorems to solve problems.

Explanation of Pre-requisites

This module will draw on general algebraic ideas including those of factor groups/rings, isomorphism theorems and ideals. Abstract algebra (MA3101) would be helpful in providing more familiarity with algebraic techniques, but is not a necessary prerequisite.

Course Description

The course explores the theme of structure in non-commutative rings and the interaction between a ring and its representations, to give a flavour of this area of algebra.

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. We start this course by exploring 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 structure of a ring gives us much information on the structure of its representations, that is, its modules. Every module is built up in some way from indecomposable modules, and so the study of the indecomposable modules is important. We show that a ring which is expressible as a direct sum of a finite number of matrix rings over division rings has only a finite number of indecomposable modules (up to isomorphism).

We then place these rings within the context of more general structural results of both rings and modules and the interactions between them. In particular we consider more general rings than matrix rings over division rings which also have a finite number of indecomposable modules. We give an overview of this area of algebra and discuss the role which these rings play within it.

Syllabus

Review of rings, ideals, homomorphisms, modules, factor modules.
Simple, semisimple and indecomposable modules.
Schur's Lemma.
Finiteness conditions on rings and modules.
The Jacobson radical.
Structure of semi-simple rings and the Wedderburn-Artin Theorem.
Krull-Schmidt theorem.
Path algebras, representation type.
Gabriel's theorem on finite representation type.

Reading list

M. Auslander, I. Reiten and S. Smalø, Representation theory of artin algebras, CUP, 1995.

T.W. Hungerford, Algebra, Springer, 1984.

N.H. McCoy, The Theory of Rings, Macmillan, 1964.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA4151 Lie Algebras Up: ModuleGuide03-04 Previous: MA4121 Projective Curves

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Last updated: 2004-02-21
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