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MA377 Rings and Modules

Lecturer: Marco Schlichting

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 lectures

Assessment: 85% by 3-hour examination 15% coursework

Prerequisites: MA106 Linear Algebra, familiarity with elementary group theory and the ring theory part of MA249 Algebra II: Groups and Rings is desirable

Leads To:

Content: A ring is an important fundamental concept in algebra and includes integers, polynomials and matrices as some of the basic examples. Ring theory has applications in number theory and geometry. A module over a ring is a generalization of vector space over a field. The study of modules over a ring R provides us with an insight into the structure of R. In this module we shall develop ring and module theory leading to the fundamental theorems of Wedderburn and some of its applications.

Aims: To realise the importance of rings and modules as central objects in algebra and to study some applications.

Objectives: By the end of the course the student should understand:

  • The importance of a ring as a fundamental object in algebra.
  • The concept of a module as a generalisation of a vector space and an Abelian group.
  • Constructions such as direct sum, product and tensor product.
  • Simple modules, Schur's lemma.
  • Semisimple modules, artinian modules, their endomorphisms. Examples.
  • Radical, simple and semisimple artinian rings. Examples.
  • The Artin-Wedderburn theorem.
  • The concept of central simple algebras, the theorems of Wedderburn and Frobenius.

Books: Recommended Reading:
Abstract Algebra by David S. Dummit, Richard M. Foote, ISBN: 0471433349
Noncommutative Algebra (Graduate Texts in Mathematics) by Benson Farb, R. Keith Dennis, ISBN: 038794057X

Additional Resources

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Year 1 regs and modules
G100 G103 GL11 G1NC

yr2.jpg
Year 2 regs and modules
G100 G103 GL11 G1NC

yr3.jpg
Year 3 regs and modules
G100 G103

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Year 4 regs and modules
G103

Archived Material
Past Exams
Core module averages