# MA377 Rings and Modules

**Lecturer: **Dr. Diane Maclagan

**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