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MC254 Algebra I

MC254 Algebra I

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 Problem Classes: 5
Tutorials: none Private Study: 52
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Explanation of Pre-requisites

This is a pure mathematics module at level 2, and thus will draw heavily on the ideas of mathematical proof and logic developed in the first year courses MC144 (Proof and logical structures) and MC145 (Algebraic structures and number systems). Fundamental mathematical concepts developed in these modules, such as sets, functions and relations from MC144, and the number systems and polynomials and their properties from MC145, will be drawn upon to develop the theory of rings and modules in MC254. The concept of a `division algorithm', which is considered for integers and polynomials in MC145, is generalised in MC254 (see also under `Course description').

Groups and rings are central concepts in MC254, and the introduction to these topics in MC145 will be very useful to students on this module. The concept of a vector space provides a motivational example of a module, so MC254 can be regarded as an extension of the theory of vector spaces developed in MC241, Linear Algebra. A good understanding of vector spaces, and the distinction between vectors and scalars, is important in understanding rings and modules.

Course Description

This course provides an introduction to the theory of rings and modules. It is a very algebraic course, and methods used will include mathematical proof and axiomatic systems, logical argument and algebraic manipulation. Consider the following (introduced in MC144, MC145 and MC241): the integers, the integers modulo n, rational numbers, real numbers, complex numbers, polynomials over a field, and matrices over a field. All of these sets have a number of features in common - they all possess an addition and a multiplication, and these operations possess a certain set of properties. Rather than study each of them one by one, we employ a central idea in mathematics -- abstraction. We define a set which has an addition and multiplication defined on it satisfying certain properties, to be a ring.

Consider also a vector space over a field ${\mathbf F}$ (which could, for example, be the real numbers). Elements of this vector space are vectors. Elements of the field are called scalars. Given a scalar $\lambda$ and a vector v, we can form a new vector $\lambda v$. Thus, the scalars `act' on the vector space. Each scalar $\lambda$ defines a map from the vector space to itself -- a vector v is mapped to $\lambda v$. This is an example of a module -- we say that the vector space is a module for ${\mathbf F}$. In this course, we consider what we can do if we allow more general sets of scalars -- i.e. we allow a ring to be the set of scalars. So a module is a `vector space' over a ring.

Much of the theory of commutative rings is motivated by the properties of the integers in addition to the usual ring properties of addition and multiplication. 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, since rings and modules form one of the important languages of advanced 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.

Further Study

Topics related to the course are covered in the material mentioned in the `Reading' section below. This module will provide students with a basic understanding of ring and module theory, one of the languages of advanced mathematics, which is employed in a number of fields of study, including group theory, number theory and linear algebra. It is also employed in the theories of algebraic groups, quantum groups and representation theory, areas of research carried out in the Department. Further study of this topic, and its relationship to field theory, is possible in the module MC382, Abstract Algebra.

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 ability to understand abstract ideas and construct rigorous logical arguments.

The ability to check that 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

Ring Theory Concepts Ring, subring, unit, division ring, quaternions, field, ring homomorphism, kernel, image, ideal, principal ideal, coprime ideals, factor ring, integral domain, divisibility in an integral domain, zero divisor, associate, irreducible element, prime element, Euclidean domain (ED), principal ideal domain (PID), unique factorisation domain (UFD).

Module Concepts Module, submodule, factor module, module homomorphism, kernel, image, cyclic module, finitely generated modules, direct sum of modules, free module.

Theorems Construction of ideals, Isomorphism theorems for rings and for modules, Every principal ideal domain is a unique factorisation domain, If R is a UFD then so is R[x]. Characterisation of a division ring as a ring with precisely two right ideals, Relationship between divisibility in an integral domain and its relation to principal ideals, Every Euclidean domain is a principal ideal domain, Every prime element in an integral domain is irreducible and the converse holds in a principal ideal domain, ${\mathbf Z}[x]$ is not a principal ideal domain, Characterisation of cyclic modules, Every finitely generated module is a homomorphic image of some Rn, Every module is a homomorphic image of a free module.

Main Examples Example of a non-principal ideal and a one-sided ideal, ideals in $\mathbf Z$, adjoining a square root to $\mathbf Z$ to get ${\mathbf Z}[\sqrt{d}]$, the norm function on ${\mathbf Z}[\sqrt{d}]$, An example of a principal ideal domain which is not a Euclidean domain, Discussion of unique factorisation in $\mathbf Z$ and ${\mathbf F}[x]$ where F is a field.

Reading list

Recommended:

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

Background:

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

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

There will be around four pieces of work set for assessment which all carry equal weight and together count for 20% of the final mark; the final examination contributes the remaining 80%.

The exam paper contains 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 carry equal weight.

Next: MC255 Algebra II Up: Year 2 Previous: MC248 Further Real Analysis

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