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Next: MC260 Mathematical Statistics Up: Year 2 Previous: MC248 Further Real Analysis

MC249 Rings and modules


MC249 Rings and modules

Credits: 10 Convenor: Dr. R. J. Marsh Semester: 2


Prerequisites: essential: MC145, MC241
Assessment: Regular Coursework: 20% One and a half hour exam: 80%

Lectures: 18 Classes: 5
Tutorials: none Private Study: 52
Labs: none Seminars: none
Project: none Other: none
Total: 75


Explanation of Pre-requisites

The definitions of a group and a ring together with the properties of polynomials are required from MC145. The concept of a vector space provides a motivational example of a module and, as such, elementary results from MC241 will be used when discussing this example.

Course Description

This course provides an introduction to the theory of rings and modules. Familiar examples of rings include the integers, the integers modulo n, rational numbers, real numbers, complex numbers, polynomials over a field, and $n\times n$ matrices over a field. The concept of a module has also been met before in the special cases of a vector space and an abelian group.

Much of the theory of commutative rings is motivated by the properties of the integers. This leads firstly to the idea of a Euclidean domain, which gives a general setting for the division algorithms already seen in MC145 for the integers and polynomials over a field, and secondly to the concept of a principal ideal domain, which gives particularly useful structure theorems for both rings and modules. Describing the structure of rings and modules is important in abstract algebra and enables us to apply this theory to many diverse areas of mathematics; from the examples listed above, it can be seen that the ideas in this course are used in number theory, linear algebra and group theory. The final part of this course considers some general theorems which allow us to describe the structure of any module.

Aims

The aim of this course is to introduce students to the basic structure and theory of rings and modules and to develop this theory to investigate important classes of integral domains and the classification of any finitely generated module as a homomorphic image of a free module. The parallels between number systems and other algebraic structures are drawn out in this course.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To understand and be able to use the main results and proofs of this course.

To be able to investigate the properties of a ring or module.

To relate the concept of an ideal to homomorphisms and factor rings.

To distinguish between the concepts of primeness and irreducibility.

To know the interrelationships between Euclidean domains, principal ideal domains and unique factorisation domains.

To understand the unique factorisation properties motivated by the example of the integers.

To understand how every finitely generated module is a homomorphic image of a free module.

Transferable Skills

The development of abstract mathematics and the axiomatic method.

The ability to apply taught principles and concepts to new situations.

The ability to present written arguments and solutions in a coherent and logical form.

The ability to use the techniques taught within the course to solve problems.

Syllabus

Ring, module, zero divisor, integral domain (ID), unit, division ring, quaternions, field, subring, submodule, ring (module) homomorphism, ideal, kernel, image, construction of ideals, principal ideal, example of a non-principal ideal and a one-sided ideal, ideals in Z, characterisation of a division ring as a ring with precisely two right ideals, coprime ideals, factor ring, factor module, isomorphism theorems.

Divisibility in an ID and its relation to principal ideals, associate, irreducible element, prime element, Z$[\sqrt{d}]$ and the norm function, Euclidean domain (ED), principal ideal domain (PID), every ED is a PID, to know that there are PIDs which are not EDs, every prime element in an ID is irreducible and the converse holds in a PID, Z[x] is not a PID, unique factorisation domain (UFD), to know that every PID is a UFD and that if R is a UFD then so is R[x], discussion of unique factorisation in Z, Z$[\sqrt{d}]$ and F[x] where F is a field.

Cyclic modules, characterisation of cyclic modules, finitely generated modules, direct sum construction for rings and modules, every finitely generated module is a homomorphic image of some Rn, free modules, every module is a homomorphic image of a free module.

Reading list

Recommended:

R. B. J. T. Allenby, Rings, Fields and Groups, 2nd Ed., Arnold.

Background:

A. W. Chatters and C. R. Hajarnavis, An Introductory Course in Commutative Algebra, Oxford University Press.

J. B. Fraleigh, A First Course in Abstract Algebra, 5th Ed., Addison-Wesley.

B. Hartley and T. Hawkes, Rings, Modules and Linear Algebra, Chapman and Hall (out of print).

W. K. Nicholson, Introduction to Abstract Algebra,, PWS-Kent.

D. Sharpe, Rings and Factorization, Cambridge University Press (out of print).

Details of Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the Summer exam period. The 20% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.


next up previous
Next: MC260 Mathematical Statistics Up: Year 2 Previous: MC248 Further Real Analysis
S. J. Ambler
11/20/1999