Introductory description

A ring is an important fundamental concept in algebra and includes integers, polynomials and matrices as some of the basic examples. Ring theory has applications in number theory and geometry. A module over a ring is a generalization of vector space over a field. The study of modules over a ring R provides us with an insight into the structure of R. In this module we shall develop ring and module theory leading to the fundamental theorems of Wedderburn and some of its applications.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Rings, algebras, division rings, classification of real division algebras, modules, free modules, Zorn's Lemma, existence of basis for modules over division rings, Wedderburn's little theorem.

Learning outcomes

By the end of the module, students should be able to:

• The importance of a ring as a fundamental object in algebra.
• The concept of a module as a generalisation of a vector space and an Abelian group.
• Constructions such as direct sum, product and tensor product.
• Simple modules, Schur's lemma.
• Semisimple modules, artinian modules, their endomorphisms. Examples.
• Radical, simple and semisimple artinian rings. Examples.
• The Artin-Wedderburn theorem.
• The concept of central simple algebras, the theorems of Wedderburn and Frobenius.

Indicative reading list

Abstract Algebra by David S. Dummit, Richard M. Foote, ISBN: 0471433349
Noncommutative Algebra (Graduate Texts in Mathematics) by Benson Farb, R. Keith Dennis, ISBN: 038794057X

Subject specific skills

Linear algebra is a key tool that under-pins many of the real world applications of mathematics. This module provides students with a far deeper conceptual understanding of linear algebra by replacing fields as the source of scalars with rings. It simultaneously introduces remarkable flexibility in allowing modules or vector spaces with potentially infinite dimension.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.