Topic outline

  • Announcements

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    • Module announcements Forum
      Here you can find all module announcements.
    • Student forum

      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.

    • Zoom link and recordings External tool

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

    • Lecture notes and textbook Folder

      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.


    • Response to Module Evaluation Results File
      121.0 KB

  • 🆕 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 corresponding lecture 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

    • Solutions for Coursework 3 File
      112.6 KB
    • Coursework 3 Assignment

      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.
    • Solutions for Coursework 2 File
      123.3 KB
    • Coursework 2 Assignment

      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.
    • Solutions for Coursework 1 File
      126.7 KB
    • Coursework 1 Assignment

      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 set: Coursework 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 7 (no lectures)

  • 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

    ACADEMIC CONTENT

    - 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.

  • Q-Review

  • Online Reading List

  • Assessment Information

    Assessment pattern: 80% final exam + 20% coursework.

    Format and dates for the in-term assessments: 3 coursework submissions, each worth 6.7% of the final mark. Submission deadlines will be roughly every 3-4 weeks. 

    Format of final assessment: 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.    

    Link to past papers: Past final exams can be found on this link. Search for "Ring Theory".

  • Final Assessment - 13th May 2024

    • Final Assessment - 13th May 2024 Quiz

      This handwritten assessment is available for a period of 3 hours and 30 minutes, within which you must submit your solutions. You may log out and in again during that time, but the countdown timer will not stop. If your attempt is still in progress at the end of your 3 and a half hours, any file you have uploaded will be automatically submitted.

      The assessment is intended to be completed within 3 hours. Please note that the additional 30 minutes is to scan and submit your answers. Please ensure that you complete the assessment within 3 hours to prevent any technical issues that may occur if you submit close to the deadline.

      After submitting, e-mail a copy to maths@qmul.ac.uk with your student number and the module code in the subject line.

      In completing this assessment:

      • You may use books and notes.

      • You may use calculators and computers, but you must show your working

      for any calculations you do.

      • You may use the Internet as a resource, but not to ask for the solution to

      an exam question or to copy any solution you find.

      • You must not seek or obtain help from anyone else.