Introductory description

This module takes the foundations of algebra learned at undergraduate level and extends them to the research level.

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.

• Group Theory
• Ring Theory
• Theory of Modules.
• Elements of Category Theory,
• Elements of Homological Algebra
• Representation Theory,
• Derived Functors.

Learning outcomes

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

• (a) surveying the range of topics from a higher more unified perspective; (b) introducing and applying categorical and homological methods, providing the language of research algebra and extending knowledge in Group Theory, Ring Theory and the Theory of Modules. (c) exposing set piece graduate topics in Group Theory, Ring Theory or the Theory of Modules.

Subject specific skills

Develop a deep understanding and applicability of the following topics:

• Free groups (with proof of existence), generators and relations.
• Tensor product and multilinear algebra
• Basics of category theory (definition of a category, functors, initial objects/universal properties, natural transformations).
• Basic homological algebra (exact sequences, homology)
• Basic noncommutative algebra (to Wedderburn’s theorem)
• Introduction to representation theory.
• Galois theory

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)