Lecturer: Christian Bohning

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 lectures

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

Formal registration prerequisites: None

Assumed knowledge: The ring theory part of the second year Maths core:

MA266 Multilinear Algebra

• Jordan normal forms
• Smith normal forms over integers
• Classification of finitely generated abelian groups

MA268 Algebra 3

• Rings
• Domains (UFD, PID, ED)
• Chinese remainder theorem
• Gauss lemma
• Eisenstein criterion

Useful background: Interest in Algebra and good working knowledge of Linear Algebra

Synergies: The following modules go well together with Rings and Modules:

• MA3G6 Commutative Algebra
• MA3E1 Groups and Representations

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA4J8 Commutative Algebra II
• MA453 Lie Algebras
• MA4M6 Category Theory
• MA4H8 Ring Theory

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