This course focuses on two important notions in abstract algebra: rings and modules. These generalise vector spaces and other concepts in linear algebra. A ring is a set equipped with two binary operations mimicking the addition and multiplication of integer, real or complex numbers. A module is an object a ring acts on – for example, modules over the integers are the same as abelian groups!

Examples of rings include square matrices, polynomials, and functions from an arbitrary set to the reals. Their applications include characterising shapes and spaces in geometry, describing many fundamental structures in mathematical physics, and designing state-of-the-art encryption algorithms in cryptography.

This Year Three module will involve the study of fundamental definitions, theorems and examples for rings and modules over rings. Students will thereby gain an overview of concepts which permeate abstract algebra and have applications throughout contemporary mathematics.

Prerequisite Modules:  MA2013 Algebra I: Groups

1. State and apply the ring axioms.
2. Understand the concepts of commutative rings, unital rings, zero divisors, integral domains, fields, units, ring homomorphisms, ring isomorphisms, subrings, ideals, principal ideals, prime ideals, maximal ideals, ring cosets, quotient rings, and the first ring isomorphism theorem.
3. Understand and work with certain classes of rings, such as matrix rings, polynomial rings, Euclidean domains, principal ideal domains and unique factorisation domains.

Understand the concept of a module over a ring, its basic properties and structures, and fundamental constructions. Classify finitely generated modules over principal ideal domains.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills:
Appreciation of the general prevalence and significance of rings and modules in mathematics. Ability to understand and apply the fundamental concepts of ring and field theory.

Transferable Skills:
Recognition of the power of mathematical abstraction and the importance of mathematical rigour. Critical thinking and problem-solving skills. Ability to present work in a scholarly manner.

1. Revision of relevant definitions and results for groups and vector spaces, from modules MA1008, MA2008 and MA2013.
2. Rings, commutative rings, unital rings, zero divisors, integral domains, fields, units, ring homomorphisms, ring isomorphisms, subrings, ideals, principal ideals, prime ideals, maximal ideals, ring cosets, quotient rings, and the first ring isomorphism theorem.
3. Classes of rings, such as matrix rings, polynomial rings, Euclidean domains, principal ideal domains and unique factorisation domains.

Modules over a ring. Simple modules. Finitely-generated modules. Morphisms of modules, their kernels and images. The first isomorphism theorem for modules. The structure theorem for finitely-generated modules over a principal ideal domain.