Department of Mathematics & Computer Science

MC254 Algebra I

Explanation of Pre-requisites

From MC144, the ideas of mathematical proof and logic, and sets and functions are needed. From MC145, number systems and polynomials and their properties, and the concepts of groups and rings are needed. The concept of ``division algorithm'' from MC145, will be generalised here. The concepts of matrices, fields, vectors spaces and subspaces and fields from MC147 and MC241 will also be needed.

Course Description

The theory of groups and rings is one of the basic languages of pure mathematics, as well as having many applications to other areas of study, such as applied mathematics, physics and chemistry. The concepts of groups and rings were met in MC145. Some familiar examples of groups are: the integers (with addition), the rational numbers (with addition), the non-zero real numbers (with multiplication), invertible matrices over a field (with multiplication), and the integers modulo (with addition). Some familiar examples of rings are the integers, the rational numbers, the real numbers, the integers modulo , and matrices over a field, all with the usual addition and multiplication.

As these number systems have many properties in common, it is good to study all of these objects at the same time, which is one of the reasons groups and rings are so useful. We begin this module by investigating the substructures of a group. For a group with a finite number of elements we have Lagrange's Theorem which helps us decide whether a subset of a group is in fact also a group in its own right. We then study the relationship between the additive groups and and so introduce the concept of a quotient group.

This leads us to the study of rings, since many properties of rings are motivated by . In MC145 we studied the division algorithm for , and we saw that the polynomials with coefficients in a field also have a division algorithm. In this module, we investigate the concept of division and study a class of rings called Euclidean domains which all have a division algorithm. Both and are Euclidean domains, but there are other new examples we study too, such as the Gaussian Integers . Not every ring shares these properties and we give a number of examples to illustrate this. The substructures of rings that play the most important role in ring theory are ideals, and we see that the ideal structure of Euclidean domains is very simple to describe. We end the course with an informal introduction to further areas of study and application. In particular MC255 leads naturally on from this module with the study of all finitely-generated abelian groups. The proofs use ring theory developed here.

Further Study

This module will provide students with a basic understanding of abstract algebra, an important language of advanced mathematics. It naturally leads on to MC255, Algebra II, in which the theories of groups and rings are developed further. Further study of rings, and the relationship to field theory, is possible in the third level module MC382, Abstract Algebra, and groups can be studied further in the third level module MC341, Group Theory. Groups and rings also play a role in a number of level modules.

Aims

The main aim of this course is to develop the knowledge of groups and rings introduced in MC145, and to introduce students to the basic structure and properties of groups and rings, as well as their substructures and quotient structures. This theory is developed to investigate Euclidean domains and principal ideal domains. 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 mo dule.

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

To understand the definition and the basic properties of groups and rings.

To investigate the structure of important examples of groups and rings.

To use Lagrange's theorem to study subgroups and quotient groups.

To understand how a factor ring can be constructed from an ideal in a ring.

To investigate the ideal structure of Euclidean domains.

Transferable Skills

The ability to understand abstract ideas and construct rigorous logical arguments.

The ability to determine whether a proof is correct.

The ability to solve mathematical problems.

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

Syllabus

Groups Binary relation, group, examples, abelian group, basic properties, cyclic group, subgroup, cyclic subgroup, order of an element, cosets, Lagrange's Theorem, index equation, conjugates, normal subgroup, factor group, group homomorphism, First Isomorphism Theorem for Groups.

Rings Ring, basic properties, subring, examples, unit, commutative ring, division ring, characterisation of a division ring as a ring with precisely two right ideals, quaternions, field, ideal, construction of ideals, principal ideal, example of a non-principal ideal and a one-sided ideal, ideals in , factor ring, zero divisor, integral domain, divisibility in an integral domain and its relation to principal ideals, Euclidean domain (ED), the Gaussian integers , principal ideal domain (PID), is not a principal ideal domain, every Euclidean domain is a principal ideal domain, an example of a principal ideal domain which is not a Euclidean domain, definition of module, abelian groups and vector spaces as modules.

Reading list

Recommended:

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

D. A. R. Wallace, Groups, Rings and Fields, Springer.

Background:

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

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

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

I. N. Herstein, Topics in algebra, Wiley.

C. R. Jordan and D. A. Jordan, Groups, Edward Arnold.

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

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

Details of Assessment

Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 2001-09-20
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.