## M3P8: Ring Theory

**Mon 12-1 in Huxley 311, Wed 10-11 in Huxley 130, Fri 10-11 in Huxley 341**

Dr. David Helm

672 Huxley

dhelm@imperial.ac.uk

Office Hours: Fri 1-3 (or by appointment)

### Course description

This course is an introduction to ring theory. The topics covered will include ideals,
factorization, the theory of field extensions, finite fields, polynomial rings in several variables,
and the theory of modules.
### Suggested References

In addition to the lecture notes, chapters 10-13 of *Algebra*, by Michael Artin, cover much of the material
we will be studying.

The course builds on the ring theory developed last year in weeks 9,10, and 11 of M2P2. Martin Liebeck has kindly made his notes
on the subject available:

Week 9

Week 10

Week 11

### Lecture Notes

I will post lecture notes on each topic we cover as the course progresses. These notes are a first draft, as this is my first time teaching this course,
so be wary of typos and other errors. If you do find any, I would very much appreciate it if you let me know!

Part 1: Basic Definitions and Examples

Part 2: Homomorphisms, Ideals, and Quotients

Part 3: Factorization

Part 4: The Chinese Remainder Theorem

Part 5: Fields and Field Extensions

Part 6: Finite Fields

Part 7: R-Modules

Part 8: Noetherian Rings and Modules

Part 9: Polynomial Rings in Several Variables

Part 10: Integral Extensions and Algebraic Integers

Part 11: Dedekind Domains

Part 12: Integers in Number Fields

Part 13: Introduction to Algebraic Geometry

### Problems

Example Sheets will be posted here every two weeks. The first two will not be assessed work, but they will be
discussed in problems classes, two weeks after being assigned. They are purely optional, but if you want feedback
you may submit them to be marked (prior to the date they are due) and I will give them back for
on the assignments.

The second two example sheets (sheets 3 and 4) WILL BE ASSESSED, and count for 5 percent of the total marks each.
The assignments will be due Mon 29 Oct, Mon 12 Nov, Mon 26 Nov, and Mon 10 Dec, at 4:00PM in the
undergraduate office.

### Mastery Material

The mastery material for fourth and fifth-year students is about modules over principal ideal domains. The relevant material
is contained in these notes. Note that there are many details
to be filled in as exercises; please feel free to contact me if you have difficulty with any of the exercises, or any other questions
about the notes. Please also let me know of any typos that you find!