The first abstract algebraic objects which are normally studied are groups, which arise naturally from the study of symmetries. The focus of this course is on rings, which generalise the kind of algebraic structure possessed by the integers: a ring has two operations, addition and multiplication, which interact in the usual way. The course begins by studying the fundamental concepts of rings (already met briefly in core Algebra): what are maps between them, when are two rings isomorphic etc. much as was done for groups. As an application, we get a general procedure for building fields, generalising the way one constructs the complex numbers from the reals. We then begin to study the question of factorization in rings, and find a class of rings, known as Unique Factorization Domains, where any element can be written uniquely as a product of prime elements generalising the case of the integers. Finally, we study modules, which roughly means we study linear algebra over certain rings rather than fields. This turns out to have powerful applications to ordinary linear algebra and to abelian groups.
Dr Richard Earl
Students should become familiar with rings and fields, and understand the structure theory of modules over a Euclidean domain along with its implications. The material underpins many later courses in algebra and number theory, and thus should give students a good background for studying these more advanced topics.
Recap on rings (not necessarily commutative or with an identity) and examples: \(\mathbb{Z}\), fields, polynomial rings (in more than one variable), matrix rings. Zero-divisors, integral domains. Units. The characteristic of a ring. Discussion of fields of fractions and their characterization (proofs non-examinable) [2]
Homomorphisms of rings. Quotient rings, ideals and the first isomorphism theorem and consequences, e.g. Chinese remainder theorem. Relation between ideals in \(R\) and \(R/I\). Prime ideals and maximal ideals, relation to fields and integral domains. Examples of ideals. Application of quotients to constructing fields by adjunction of elements; examples to include \(\mathbb{C} = \mathbb{R}[x]/ (x^2 + 1)\) and some finite fields. Degree of a field extension, the tower law. [4]
Euclidean Domains. Examples. Principal Ideal Domains. EDs are PIDs. Unique factorisation for PIDs. Gauss's Lemma and Eisenstein's Criterion for irreducibility. [3]
Modules: Definition and examples: vector spaces, abelian groups, vector spaces with an endomorphism. Submodules and quotient modules and direct sums. The first isomorphism theorem. [2]
Row and column operations on matrices over a ring. Equivalence of matrices. Smith Normal form of matrices over a Euclidean Domain. [1.5]
Free modules and presentations of finitely generated modules. Structure of finitely generated modules of a Euclidean domain. [2]
Application to rational canonical form and Jordan normal form for matrices, and structure of finitely generated Abelian groups. [1.5]