# NONCOMMUTATIVE ALGEBRA - 2020/1

Module code: MATM049

## Module Overview

This module extends students’ knowledge of ring theory and introduces various topics in noncommutative algebra.

### Module provider

Mathematics

FISHER David (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 117

Lecture Hours: 33

Semester 2

## Prerequisites / Co-requisites

MAT2048 Groups and Rings

## Module content

Indicative content includes:

• Finite-dimensional algebras.

• Lie algebras, subalgebras, ideals, quotient algebras and direct sums.

• Derivations, homomorphisms and automorphisms of algebras.

• Modules over rings; vector spaces over division rings.

• Noetherian and Artinian rings.

• Wedderburn's decomposition theorem.

• Nilpotency, solvability and semisimplicity.

• Tensor products of rings and modules. Application to algebra automorphisms.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 20
Examination EXAMINATION (2 HOURS) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate their ability to

• construct and interpret mathematical arguments in the context of this module;

• display subject knowledge by recalling key definitions and results;

• apply the techniques learnt to both routine and unfamiliar problems.

Thus, the summative assessment for this module consists of:

• One two-hour examination at the end of Semester 2, worth 80% of the module mark.

• In-semester test, worth 20% of the module mark

Formative assessment and feedback

Students receive written comments on their marked test and assignments.  Verbal feedback is provided in lectures and office hours.

## Module aims

• enhance students' appreciation of abstract algebraic structure theory,
• develop an understanding of noncommutative ring theory and modules over rings,
• provide a foundation for independent study of algebraic topics.

## Learning outcomes

 Attributes Developed 001 Know the definitions and language associated with rings, algebras and modules and be familiar with examples K 002 Appreciate the relationships between the structures studied in this module and those encountered in previous algebraic modules KCT 003 Know and be able to apply a variety of standard results and structure theorems, as listed in the module content KC 004 Devise solutions of algebraic problems involving the concepts studied KC 005 Read and interpret varying presentations of the material CT 006 Construct simple proofs similar to those encountered in the module KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• An enhanced awareness of the theory and applications of abstract algebra

• Experience of the methods used to interpret, understand and solve problems in the topics covered.

The learning and teaching methods include:

• Three 50-minute lectures per week for eleven weeks, some being used as tutorials, problem classes and a test.

• Online notes supplemented by additional examples in lectures.

• Unassessed coursework consisting of exercises from the notes.

• Personal assistance given to individuals and small groups in office hours.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.