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MA2111 Algebra I

MA2111 Algebra I

Credits: 10 Convenor: Dr. A. E. Henke Semester: 1 (weeks 7 to 12)
Prerequisites: desirable: MA1102, MA2102
Assessment: Regular coursework: 20% One and a half hour exam: 80%
Lectures: 18 Problem Classes: 4
Tutorials: none Private Study: 46
Labs: 3 Seminars: none
Project: none Other: none
Surgeries: 4 Total: 75

Subject Knowledge

Aims

The main aim of this course is to develop the knowledge of groups and rings introduced in MA1102, 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.

Learning Outcomes

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.
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.

Methods

Class sessions, surgeries and laboratories together with some handouts.

Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination in January. The 20% coursework contribution will be determined by students' solutions to four sets of work. The exam paper will contain 4 questions. Any number of questions may be attempted, but only the best 3 answers will be taken into account. Full marks may be obtained for answers to 3 questions. All questions will carry equal weight.

Subject Skills

Aims

Develop written communication skills and problem solving skills.

Learning Outcomes

Students will be able to use the techniques taught within the module to solve problems, and be able to present arguments and solutions in a coherent and logical form.

Methods

Class sessions, surgeries, laboratories.

Assessment

Marked problem sheets, examination.

Explanation of Pre-requisites

From MA1101, the ideas of mathematical proof and logic, and sets and functions are needed. From MA1102, number systems and polynomials and their properties, and the concepts of groups and rings are needed. The concept of ``division algorithm'' from MA1102, will be generalised here. The concepts of matrices, fields, vectors spaces and subspaces and fields from MA1152 and MA2102 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 MA1102. Some familiar examples of groups are: the integers (with addition), the rational numbers (with addition), the non-zero real numbers (with multiplication), invertible $n\times n$ matrices over a field (with multiplication), and the integers modulo $n$ (with addition). Some familiar examples of rings are the integers, the rational numbers, the real numbers, the integers modulo $n$, 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 $\bf Z$ and ${\bf Z}_n$ 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 $\bf Z$. In MA1102 we studied the division algorithm for $\bf Z$, 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 $\bf Z$ and ${\bf F}[x]$ are Euclidean domains, but there are other new examples we study too, such as the Gaussian Integers ${\bf Z}[i]$. 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 MA2161 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 MA2161, 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 MA3101, Abstract Algebra, and groups can be studied further in the third level module MA3131, Group Theory. Groups and rings also play a role in a number of level $4$ modules.

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 $\mathbf Z$, factor ring, zero divisor, integral domain, divisibility in an integral domain and its relation to principal ideals, Euclidean domain (ED), the Gaussian integers ${\bf Z}[i]$, principal ideal domain (PID), ${\mathbf Z}[x]$ 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).

Resources

Problem sheets and additional handouts.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA2121 Linear Analysis Up: ModuleGuide03-04 Previous: MA2102 Linear Algebra

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Last updated: 2004-02-21
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