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MC440 Commutative Algebra

MC440 Commutative Algebra

Credits: 20 Convenor: Dr. N. J. Snashall Semester: 2

Prerequisites: essential: MC241, MC242, MC382 desirable:
Assessment: Coursework: 10% Three hour exam: 90%

Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

The definition and basic properties of a group including the construction of the factor group are assumed from MC242; and from its first year prerequisites we use the definition of a ring and properties of polynomials. 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

Commutative algebra is a beautiful subject in its own right but is also important for the study of algebraic geometry (and the study of polynomial rings in several variables) and algebraic number theory (and the study of the integers). Familiar examples of commutative rings include the integers, the integers modulo n, rational numbers, real numbers, complex numbers and polynomials 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. The course begins with some of the standard theory concerning commutative rings and modules, building on that already covered in MC382.

This course concentrates on the problems of factorisation and its applications to algebraic number theory. It is well known that every positive integer can be uniquely factorised as a product of primes. A prime integer is irreducible, that is, it cannot be written as a product of two strictly smaller positive integers. However, there are number systems where an element may have two distinct factorisations as a product of irreducible elements and thus factorisations are not necessarily unique, nor need they even exist! We investigate such questions and see that where there is no unique factorisation of elements then, by studying particular sets of elements, called ideals, we may nevertheless have unique factorisation of ideals. This enables us to draw parallels between algebraic and number theoretic concepts.

Aims

This module aims to develop an understanding of commutative algebra and to indicate its applications to other areas of mathematics. To this end the concepts of unique factorisation of both elements and ideals, chain conditions and Dedekind domains are studied together with their relevance to algebraic number theory.

Objectives

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

To understand, reconstruct and apply the main results and proofs covered in this course.

To determine the properties of a ring or module and be able to investigate the ideal structure of a commutative ring.

To be able to investigate problems concerning existence and uniqueness of factorisations.

To understand the relation between the different classes of rings introduced in this course.

To use the concepts and results taught in this module to solve a variety of problems.

Transferable Skills

The development of understanding of the abstract method within commutative algebra.

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

Commutative ring, module, integral domain (ID), idempotent, nilpotent element, ideal, principal ideal, polynomial ring, power series ring, ring (module) homomorphism, factor ring (module), radical of an ideal, prime ideal, maximal ideal, isomorphism theorems, characterisation of prime and maximal ideals.

Multiplicatively closed set, ring of fractions, localisation at a prime ideal, module of fractions, local ring, local property.

Division in an ID, irreducible element, prime element, norm function for Z$[\sqrt d]$, Euclidean domain (ED), principal ideal domain (PID), unique factorisation domain (UFD), every prime element is irreducible and the converse holds in a UFD, every ED is a PID, every PID is a UFD.

Cyclic module, annihilator of an element of a module, finitely-generated module and ideal, generating set, acc and Max, Noetherian module, Noetherian ring, if R is a Noetherian ring then both S-1R and R[x] are Noetherian rings, characterisation theorem for cyclic modules, characterisation theorems for Noetherian modules, dcc and Min, Artinian ring, if R is an Artinian ring then every prime ideal is maximal and R has only a finite number of maximal ideals.

Primary ideal, primary decomposition and minimal primary decomposition, irreducible ideal, Dedekind domain, characterisations of primary ideals and radicals, every proper ideal of a commutative Noetherian ring has a primary decomposition, every non-zero proper ideal of a Noetherian ID in which every non-zero prime ideal is maximal can be uniquely expressed as a product of primary ideals whose radicals are all distinct, every PID is a Dedekind domain, first and second uniqueness theorems for primary decompositions.

Reading list

Recommended:

B. Hartley and T. Hawkes, Rings, Modules and Linear Algebra, Chapman and Hall.

R. Y. Sharp, Steps in Commutative Algebra, LMS Student Texts 19, CUP.

Background:

M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley (out of print).

P. M. Cohn, Algebra vol.1, Wiley.

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

I. N. Herstein, Topics in Algebra, 2nd edition, Wiley.

N. Jacobson, Basic Algebra vol.1, Freeman.

H. Matsumura, Commutative Ring Theory, CUP.

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

Details of Assessment

There will be at least five pieces of work set for assessment which together count for 10% of the final mark.

There are eight questions on the examination paper; any number of questions may be attempted but full marks may be gained from answers to five questions; all questions carry equal weight.

Next: MC446 Algebraic Topology Up: Year 4 Previous: MC430 Approximation Theory

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