# MA377 Rings and Modules

**Lecturer: **Dmitriy Rumynin

**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:

MA251 Algebra I: Advanced Linear Algebra:

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

MA249 Algebra II: Groups and Rings:

- 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:

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

**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