# MSc Pure Mathematics

Year of entry: 2024

## Course unit details:Noncommutative Algebra

Unit code MATH62042 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 2 No

### Overview

Nature is inherently noncommutative---just try putting on your shoes and socks in the wrong order---and noncommutative structures are increasingly important throughout mathematics and physics. In this course, we will examine in detail some of the most important noncommutative algebras that appear "in nature" and prove some of the basic structure theorems about noncommutative rings. One of the most fundamental such algebras is the quaternions---if you ignore the fact that it is not commutative, then this is a field that is 4-dimensional as a real vector space! Famously when Hamilton discovered it in the 19-th century he carved its formulae on a bridge lest he forget them. Another is the Weyl algebra---sometimes called the "algebra of quantum mechanics" since its structure encodes the Uncertainty Principle, and sometimes called a ring of differential operators since it encodes the algebraic aspects of differential equations.

### Pre/co-requisites

Students are not permitted to take, for credit, MATH42041 in an undergraduate programme and then MATH62041 in a postgraduate programme at the University of Manchester, as the courses are identical.

### Aims

To introduce students to noncommutative algebra.

### Learning outcomes

On successful completion of this course unit students will be able to:

• demonstrate familiarity with fundamental noncommutative algebras, in particular the quaternions, matrix rings over division rings and the Weyl algebras;
• produce proofs involving the basic concepts of modules as covered in the course unit, in particular annihilators, the artinian and noetherian conditions, homomorphisms and endomorphism rings;
• produce proofs similar to parts of the proofs of the basic structure theorems in noncommutative algebra established in the course, in particular the Wedderburn density theorem, the classification of simple Artinian rings and the structure of modules over principal ideal domains;
• analyse examples using those structure theorem.

### Syllabus

• Introduction and preliminaries: fields, rings and matrices;
• Examples of noncommutative rings, the quaternions and the Weyl algebra;
• Wedderburn-Artin theory - many rings are matrices;
• The structure of Artinian rings;
• Introduction to representation theory;
• The structure of modules over principal ideal domains;
• Division rings.

### Assessment methods

Method Weight
Other 20%
Written assignment (inc essay) 80%
• Coursework: weighting 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Cohn, P. M. Introduction to ring theory. Springer-Verlag, 2000.
• Passman, D. S. A course in ring theory. The Wadsworth & Brooks/Cole Mathematics Series, 1991.
• Lam, T. Y. A first course in noncommutative rings. Second edition. Graduate Texts in Mathematics, 131. Springer-Verlag, New York, 2001

### Study hours

Scheduled activity hours
Lectures 11
Tutorials 11
Independent study hours
Independent study 128

### Teaching staff

Staff member Role
Cesare Giulio Ardito Unit coordinator
Charles Eaton Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.