MAGIC073: Commutative Algebra

Course details

A core MAGIC course

Semester

Autumn 2026
Monday, October 5th to Friday, December 11th

Hours

Live lecture hours
10
Recorded lecture hours
10
Total advised study hours
80

Timetable

Thursdays
11:05 - 11:55 (UK)

Course forum

Visit the MAGIC073 forum

Description

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 fundamentals of the subject.
 
In this course, the first part presents the essential concepts of commutative algebra, while the remaining parts 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 chapter.  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 are available on the MAGIC website from the start 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 assessment will follow in due time.

Accessibility: the notes are available in both PDF and HTML formats.


Prerequisites

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. 

Syllabus

  1. I Factorization and divisibility

    1. 1 A story about numbers

    2. 2 Elements of commutative rings

    3. 3 Factorization

    4. 4 Division with remainder

    5. 5 Principal, prime and maximal ideals

  2. II A structural approach

    1. 6 Localization

    2. 7 Polynomials and their solutions (algebraically)

    3. 8 Modules

    4. 9 Factorization into ideals

  3. III Solving polynomial equations

    1. 10 A story about an equation

    2. 11 Shape without coordinates

    3. 12 Polynomials and their solutions (geometrically)

    4. 13 Local properties of curves

Lecturer

  • Professor Jan Grabowski

    Professor Jan Grabowski

    University
    University of Lancaster

Bibliography

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

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

Assessment

The assessment for this course will be released on Monday 11th January 2027 at 00:00 and is due in before Friday 22nd January 2027 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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