Rings MA416

Welcome to the homepage of Rings (MA416), a 5-ECTS module in Semester 1 2024-2025. Rings are (roughly) sets of mathematical objects that can be added, subtracted and multiplied together. Examples include the integers, the rational numbers, the real numbers, the complex numbers, the set of all 3 by 3 matrices with integer entries, the set of all polynomials in one variable with rational coefficients, the set of all continuous real-valued functions defined on the real number, the set of all subsets of any set (with specified operations), and many more.

People have been studying specific examples of rings for
thousands of years, but the emergence of ring theory as a branch of
mathematics in its own right is a recent development. Much of the
activity that led to the modern formulation of ring theory took place
in the first half of the 20th century. Ring theory is powerful in
terms of its scope and generality, and can be simply described as
the study of systems in which addition and multiplication are possible, and satisfy certain (sensible) rules about how they behave and interact.

Course activities (what students are expected to do)

  1. Study the course text, which is the set of lecture notes posted below.
  2. Participate in the lectures, which will take place on Tuesdays at 11.00 and Wednesdays at 12.00, all in ADB-1020. Lectures will run for 12 weeks, starting in the week of September 9.
  3. Complete three homework assignments, with due dates in Weeks 5, 8 and 11.

Lecture Notes

This set of lecture notes is the text for the course. The lectures will be used to discuss their content, but not to go through every detail. Studying the text and participating in the lectures are both important, they are not alternatives to each other.

Chapter 1: What is a Ring?
Chapter 2: Polynomial Rings and Factorization
  • Lecture 6 Polynomial Rings
  • Lecture 7 Division in a polynomial ring over a field
  • Lecture 8 Reducible and irreducible polynomials
  • Lecture 9 Irreducibility of rational (and integer) polynomials

Chapter 3: Ideals, Homomorphisms and Factor Rings

Assessment

Assessment will consist of the following elements:

  • Three homework assignments, with due dates on Wednesday October 9 (Week 5), Wednesday October 30 (Week 8), Wednesday November 20 (Week 11). Submission will be via the MA416 Canvas page.
  • A formal exam in the winter exam session in December.
  • Ten (optional) weekly challenges.

Homework Sheets

Weekly Challenges