This is the home page for Rings & modules, given in the first semester of 2010–11 at Queen Mary, University of London. This module is being rested for the academic year 2011–12, but the information here may be useful.
Background to the course
The notion of a ring is an important one in algebra, since it generalises important examples such as the integers, fields, polynomial rings and matrix rings, and has applications to geometry, number theory and topology. Rings are often understood through modules; these are spaces on which a ring acts, generalising the idea of a vector space.
This module will define some basic notions of ring theory, and then explore some further topics.
Prerequisites
It will be useful to have done an abstract algebra course and to have seen a definition of a ring before, but don't worry too much about this. In fact, the definition of a ring varies according to what you're using it for. What will be really helpful for this course is to know basic linear algebra, since vector spaces will be used as an important example throughout the course. So you should make sure that you're really comfortable with bases, dimension and linear maps.
Syllabus
This course is designed to give an overview of rings and modules. The definition of a ring will not be assumed, but familiarity with the basic definitions will be helpful. What is essential is a thorough grounding in linear algebra.
Topics
- Definitions of rings, modules, homomorphisms. The Isomorphism Theorems.
- Inverses and division rings. Wedderburn's theorem on finite division rings.
- Direct sums of rings and modules; central idempotents.
- Semisimple modules. The Artin-Wedderburn Theorem on semisimple rings.
- Chain conditions, Artinian and Noetherian rings and modules. Composition series.
- The Jacobson radical and Artinian rings. Hopkins's Theorem.
Learning outcomes
By the end of the course, a student should be able to:
- define rings, modules, ideals amnd homomorphisms, and prove the Isomorphism Theorems;
- explain inverses and division rings, and prove Wedderburn's theorem on finite division rings;
- define direct sums and coproducts of modules and direct sums of rings, and explain the relationship between direct sums and central idempotents;
- define and give essential properties of semisimple rings and modules, and prove the Artin-Wedderburn Theorem;
- define and give basic properties of artinian and noetherian modules, and prove the Jordan-Hölder Theorem;
- define and give basic properties of the Jacobson radical, and prove Hopkins's Theorem.
The exam
Rubric
The exam rubric for this module in recent years has been: you may attempt as many questions as you wish, and except for the award of a bare pass, only the best four attempts count.
Some past questions
A couple of comments on these past papers:
- The syllabus changed between 2009 and 2010, so only the 2010 and 2011 papers reflect the current syllabus. Some of the questions on the other papers are useful revision, and it should be clear which questions are inappropriate.
- In 2007 and 2008 there was a different lecturer, so the style (and perhaps the difficulty) of the exam is rather different. So for revision purposes the 2005 and 2006 papers might be more appropriate.