## Topic outline

• ### Announcements

• Here you can find all module announcements.
• This forum is available for everyone to post messages to. You should ask all non-personal questions using this forum. Feel free to reply to other students if you are able to.

• Recordings of the lectures can be found at this link under the "Cloud Recordings" tab.

• Here you can access the handwritten notes from all lectures. The material is directly based on Chapter 2 of the following book, completely available in the Online Reading List:

• Peter J. Cameron, Introduction to Algebra (Second Edition), Oxford University Press, 2008.

• ### 🆕 Final Exam Information

Format of the final exam: Your final examination will be online. It will be 3 hours in duration with SpLD accommodations handled separately. As you will have access to the textbook and your notes during the test, the exam questions will not ask you to repeat exactly some definition or proof we have done in the lectures -- instead, the questions will test your understanding of the material and how you can apply it in new contexts.

Sample exam: The exam will be similar in structure and style to last year's exam.

Solutions to sample exam: Full solutions to the sample exam were provided in the review lecture. A recording of the lecture is linked below, together with the correspondinlecture notes.

Link to other past papers: Other past final exams can be found on this link. Search for "Ring Theory".
• ### Week 1

#### TOPICS

1. Definition of rings: textbook Section 2.1.
2. Examples of rings: textbook Section 2.2.
3. Basic properties of rings: textbook Section 2.3.

#### WEEK ACTIVITIES

1. Familiarise yourself with this QMplus page.
2. Attend the Monday and Friday lectures.
3. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
4. Non-examinable. Read this short article about the history of Ring Theory and why mathematicians developed it:
• ### Week 2

#### TOPICS

1. Basic properties of rings: textbook Section 2.3.
2. Subrings: textbook Section 2.4.
3. Matrix rings: textbook Exercise 2.2.

#### WEEK ACTIVITIES

1. Attend the Monday and Friday lectures.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
3. Work on Coursework 1, due Sun Feb 18 at 11:59pm.
• ### Coursework assignments and submission

• You can find the coursework assignment following this link. This coursework is worth 6.7% of your final module mark.

Please submit your solutions as a single pdf file (either typed in LaTeX or handwritten and scanned) before Sunday Apr 14 at 11:59pm. Solutions will be posted soon afterwards.
• You can find the coursework assignment following this link. This coursework is worth 6.7% of your final module mark.

Please submit your solutions as a single pdf file (either typed in LaTeX or handwritten and scanned) before Sunday Mar 17 at 11:59pm. Solutions will be posted soon afterwards.
• You can find the coursework assignment following this link. This coursework is worth 6.6% of your final module mark.

Please submit your solutions as a single pdf file (either typed in LaTeX or handwritten and scanned) before Sunday Feb 18 at 11:59pm. Solutions will be posted soon afterwards.
• ### Week 3

#### TOPICS

1. The ring of all subsets of a setCoursework 1, Exercise 1b.
2. Review of equivalence relations and partitions: textbook Section 1.18.
3. Cosets: textbook Section 2.5.
4. Homomorphisms: textbook Section 2.6.

#### WEEK ACTIVITIES

1. Attend the Monday and Friday lectures.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
3. Work on Coursework 1, due Sun Feb 18 at 11:59pm.
• ### Module Description

This module is about the theory of rings, continuing from the basics taught in Introduction to Algebra. We will revise the definition of a ring as a set with addition and multiplication operations satisfying some natural rules, inspired by examples arising all over mathematics, such as rings of integers, polynomials, and matrices. We will introduce and study special classes of rings (e.g fields) and related algebraic structures, in a way that mirrors the approach to Group theory in MTH6106. Familiar arithmetic concepts such as the Euclidean algorithm will be seen as special cases of much more general results about rings. Another central theme of the module is how different types of rings sit in relation to each other.

Throughout this module there is a strong emphasis on abstract thinking and proof.

• ### Week 4

#### TOPICS

1. Homomorphisms, images and kernels: textbook Section 2.6.
2. Ideals, ideal test: textbook Section 2.6.
3. Factor rings: textbook Section 2.8.

#### WEEK ACTIVITIES

1. Attend the Monday and Friday lectures.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
3. Work on Coursework 1, due Sun Feb 18 at 11:59pm.
• ### Week 5

#### TOPICS

1. Factor rings: textbook Section 2.8.
2. Isomorphism theorems: textbook Section 2.8.

#### WEEK ACTIVITIES

1. Attend the Monday and Friday lectures.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
3. Work on Coursework 2, due Sun Mar 17 at 11:59pm.
• ### Week 6

#### TOPICS

1. Zero-divisors, integral domains: textbook Section 2.10.
2. Polynomials: textbook Section 2.9.
3. Units, associate classes: textbook Section 2.10.

#### WEEK ACTIVITIES

1. Attend the Monday and Friday lectures.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
3. Work on Coursework 2, due Sun Mar 17 at 11:59pm.

• ### Week 8

#### TOPICS

1. Irreducible elements, unique factorisation domains: textbook Section 2.11.
2. Greatest common divisors: textbook Section 2.11.

#### WEEK ACTIVITIES

1. Attend the Monday and Friday lectures.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
3. Work on Coursework 2, due Sun Mar 17 at 11:59pm.
• ### Week 9

#### TOPICS

1. Greatest common divisors in unique factorisation domains: textbook Section 2.11.
2. Generating ideals in a domain: textbook Section 2.12.
3. Principal ideal domains: textbook Section 2.12.
4. Euclidean domains: textbook Section 2.13.

#### WEEK ACTIVITIES

1. Attend the Monday and Friday lectures.
2. 🆕 Attend the extra lecture on Thursday 14:00-16:00 in Queens Building LG2.
3. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Friday.
• ### Week 10

#### TOPICS

1. Euclidean domains: textbook Section 2.13.
2. Maximal ideals and fields: textbook Section 2.15.

#### WEEK ACTIVITIES

1. Attend the Monday lecture.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Fridays.
3. Work on Coursework 3, due Sun Apr 14 at 11:59pm.
• ### Week 11

#### TOPICS

1. Maximal ideals and fields: textbook Section 2.15.
2. Field extensions: textbook Section 2.16.

#### WEEK ACTIVITIES

1. Attend the Friday lecture.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Fridays.
3. Work on Coursework 3, due Sun Apr 14 at 11:59pm.
4. Non-examinable. If you are interested in learning more about Gaussian integers, their factorisation, and some interesting applications to number theory, I recommend taking a look at the following survey article:
• ### Syllabus

1. Definition of a ring, simple properties, examples.
2. Subrings, subring test. Cosets.
3. Ideals. Construction of factor rings.
4. Homomorphisms. Image and kernel.
5. Correspondence and isomorphism theorems.
6. Integral domains, polynomial rings, zero divisors, units, groups of units, examples.
7. Unique factorisation domains, principal ideal domains, Euclidean domains, fields and maximal ideals.
8. Finite fields.
9. Field of fractions.

• ### Module Aims and Learning Outcomes

- Demonstrate understanding of the definition and basic properties of rings, subrings, ring homomorphisms, ideals, and factor rings.
- Demonstrate understanding of the hierarchy of integral domains: unique factorisation domains, principal ideal domains, Euclidean domains.
- Demonstrate understanding of the basic theory of fields, finite fields, field extensions.

#### DISCIPLINARY SKILLS

- Describe the different axioms satisfied by rings, integral domains, and fields, and determine if a given algebraic structure satisfies them.
- Explain how various basic properties of rings follow from the axioms, by providing general proofs of them.
- State standard results in basic ring theory, and apply these results in concrete examples.
- Calculate in various types of rings, such as polynomial rings, Gaussian integers, and simple factor rings.
- Explain how general results in ring theory can be applied to solve problems in related areas, such as number theory and classical algebra.

#### ATTRIBUTES

- Construct and develop mathematical arguments in a clear and rigorous way.
- Demonstrate knowledge key mathematical concepts by illustrating them in a general framework and applying them to the solution of concrete problems.

• ### Feedback and Where to Get Help

There will be numerous ways for you to get feedback on your progress: You will receive written feedback on all your coursework submissions; in addition, official coursework solutions will be provided. You will be able to ask questions in the forum, which will be used to guide the content of the seminar sessions on Fridays. Weekly learning support hours will be provided as well.

Questions about the content of the module or its organisation should be posted in the student forum, which will be regularly monitored by the module leader. You are also encouraged to attend the extra learning support hour. For questions regarding personal matters, you can send an email directly to the module leader.

• ### Week 12

#### TOPICS

1. Field extensions, finite fields: textbook Section 2.16.

#### WEEK ACTIVITIES

1. Attend the Monday lecture.
2. Review the material on your own and write down any questions you have. Post them on the student forum so that I can answer them and/or discuss them in the second hour on Fridays.
3. Work on Coursework 3, due Sun Apr 14 at 11:59pm.
4. Attend the Friday review lecture.