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MA22017: Groups and rings

[Page last updated: 03 June 2024]

Academic Year: 2024/25
Owning Department/School: Department of Mathematical Sciences
Credits: 10 [equivalent to 20 CATS credits]
Notional Study Hours: 200
Level: Intermediate (FHEQ level 5)
Period:
Semester 2
Assessment Summary: EXCB 100%
Assessment Detail:
  • Closed-book written examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA22020
Learning Outcomes: By the end of this unit, you will be able to:
  • Demonstrate understanding of the basic theory of groups and rings.
  • Factorise in various integral domains you have met throughout the course and demonstrate understanding of the general theory.
  • Apply the Orbit-Stabiliser theorem for groups and work with the First Isomorphism Theorem and the Orbit Partition Theorem.



Synopsis: You will explore the basic algebraic structures of groups and rings, met briefly in the earlier algebra courses. As in Advanced Linear Algebra, the unit combines the development of structure using axioms with applications (for example, groups to geometry, and rings to polynomials), leading to some of the jewels of basic algebra, such as the study of Platonic solids and unique factorisation in polynomial rings.

Content: Elementary axiomatic theory of groups and rings. Subgroups, normal subgroups and quotient groups. Group homomorphisms and isomorphisms. Order of an element and a group. Cyclic groups. Actions, orbits and stabilisers. Cayley's theorem. Orbit partition theorem. Orbit stabiliser theorem. Integral domains, fields, characteristic. Subrings and products of rings. Homomorphisms, ideals and quotient rings. Modules. Isomorphism theorems. Fields of fractions. Polynomial rings. Maximal ideals and prime ideals. Factorisation in integral domains. Unique factorisation in principal ideal domains. Eisenstein criterion and other criteria for factorisation in polynomial rings. Modules over principal ideal domains. Further topics to me chosen from: Tensor products. Quaternions. Normed algebras.

Course availability:

MA22017 is Compulsory on the following courses:

Department of Computer Science
  • USCM-AFM32 : MComp(Hons) Computer Science and Mathematics (Year 2)
  • USCM-AKM32 : MComp(Hons) Computer Science and Mathematics with professional placement (Year 2)
  • USCM-AKM32 : MComp(Hons) Computer Science and Mathematics with study abroad (Year 2)

MA22017 is Optional on the following courses:

Department of Computer Science
  • USCM-AFB32 : BSc(Hons) Computer Science and Mathematics (Year 2)
  • USCM-AKB32 : BSc(Hons) Computer Science and Mathematics with professional placement (Year 2)
  • USCM-AKB32 : BSc(Hons) Computer Science and Mathematics with study abroad (Year 2)
Department of Economics
  • UHES-AFB12 : BSc(Hons) Economics and Mathematics (Year 2)
  • UHES-AKB12 : BSc(Hons) Economics and Mathematics with professional placement (Year 2)
  • UHES-AKB12 : BSc(Hons) Economics and Mathematics with professional placement and study abroad (Year 2)
  • UHES-AKB12 : BSc(Hons) Economics and Mathematics with study abroad (Year 2)
Department of Mathematical Sciences
  • USMA-AFB30 : BSc(Hons) Mathematics (Year 2)
  • USMA-AFB32 : BSc(Hons) Mathematics and Statistics (Year 2)
  • USMA-AKB32 : BSc(Hons) Mathematics and Statistics with professional placement (Year 2)
  • USMA-AKB32 : BSc(Hons) Mathematics and Statistics with study abroad (Year 2)
  • USMA-AFB33 : BSc(Hons) Mathematics, Statistics and Data Science (Year 2)
  • USMA-AKB33 : BSc(Hons) Mathematics, Statistics and Data Science with professional placement (Year 2)
  • USMA-AKB33 : BSc(Hons) Mathematics, Statistics and Data Science with study abroad (Year 2)
  • USMA-AKB30 : BSc(Hons) Mathematics with professional placement (Year 2)
  • USMA-AKB30 : BSc(Hons) Mathematics with study abroad (Year 2)
  • USMA-AFM30 : MMath(Hons) Mathematics (Year 2)
  • USMA-AKM30 : MMath(Hons) Mathematics with professional placement (Year 2)
  • USMA-AKM31 : MMath(Hons) Mathematics with study abroad (Year 2)
Department of Physics
  • USPH-AFB26 : BSc(Hons) Mathematics and Physics (Year 2)
  • USPH-AKB26 : BSc(Hons) Mathematics and Physics with professional placement (Year 2)
  • USPH-AKB26 : BSc(Hons) Mathematics and Physics with study abroad (Year 2)
  • USPH-AFM26 : MSci(Hons) Mathematics and Physics (Year 2)
  • USPH-AKM26 : MSci(Hons) Mathematics and Physics with professional placement (Year 2)
  • USPH-AKM26 : MSci(Hons) Mathematics and Physics with study abroad (Year 2)

Notes:

  • This unit catalogue is applicable for the 2024/25 academic year only. Students continuing their studies into 2025/26 and beyond should not assume that this unit will be available in future years in the format displayed here for 2024/25.
  • Courses and units are subject to change in accordance with normal University procedures.
  • Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
  • Find out more about these and other important University terms and conditions here.