MAGIC073: Commutative Algebra

This course in commutative algebra is aimed at non-experts. The pre-requisites are those of standard undergraduate courses in abstract algebra and linear algebra. Commutative algebra has wide-ranging applications in pure maths, and the objective of the course is to learn the fundaments of the subject.

In this course, the first two sections present the essential concepts of commutative algebra, while the remaining sections present topics which may not have been seen in an undergraduate introductory course on commutative algebra. The content of the notes has been selected from the material in the references at the end, and students are encouraged to consult these textbooks if they want to explore some topics in greater depth, or their applications to a broad range of research areas.

Within the lecture notes, there are several exercises at the end of each section. Every student is encouraged to attempt as many exercises as they can, including from the selected bibliography.

Your lecturer will recommend some exercises, mostly taken from the notes. Model solutions of these will be made available on the MAGIC website by the end of the course. Every student is assumed to understand the importance of engaging with the practical aspects of a course.

The entire material covered during the lectures is examinable, including the exercises. More details about the assessmet will follow in due time.

For the computer-algebra enthusiasts, there is reference to a resource which may be of interest, and you are also recommended to consider the numerous possibilities offered by MAGMA (http://magma.maths.usyd.edu.au/magma/) and other software such as Python or SageMath (https://www.sagemath.org/). No assessment will require the use of such software, but, unless specified, it is not forbidden to use computing tools.

Accessibility: please contact your lecturer if you need an alternative format for the lecture notes and exercises.

The pre-requisites are those of standard undergraduate courses in abstract algebra and linear algebra. It may help to have some background in commutative rings and ideal theory, but this is not essential.

1. Commutative algebra: the essentials (rings, ideals, homomorphisms, localisation)
2. Modules
3. Integral dependence
4. Prime and maximal ideal spectra
5. A brief taste of algebraic geometry: algebraic sets and Hilbert's Nullstellensatz
6. Primary decomposition
7. Dimension in commutative rings.

MAGIC073: Commutative Algebra

Course details

Semester

Hours

Timetable

Description

Prerequisites

Syllabus

Lecturer

Professor Jan Grabowski

Bibliography

Follow the link for a book to take you to the relevant Google Book Search page

Assessment

Files

Lectures