Basic concepts of ring theory: subrings, ideals, quotient, product, matrix and polynomial rings; factorisation in integral (euclidean, principal ideal) domains. Basic concepts of module theory: submodules, quotient modules, direct sums, homomorphisms, finitely generated, cyclic, free and torsion modules, annihilator ideals. Matrices and finitely generated modules over a principal ideal domain: Equivalence of matrices, structure theory of modules, applications to abelian groups and to vector spaces with a linear transformation.
4CCM121A/5CCM121B Introduction to Abstract Algebra and 4CCM114A/5CCM114B Linear Algebra and Geometry II
Semester 1 only students will be set an alternative assessment in lieu of in-person exams in January.Full year students will complete the standard assessment.
Educational aims & objectives
This module aims to develop the general theory of rings (especially commutative ones) and then study in some detail a new concept, that of a module over a ring. Both abelian groups and vector spaces may be viewed as modules and important structure theorems for both follow from the general theory. The theory of rings and modules is key to many more advanced algebra courses e.g. Algebraic Number Theory. It can also help with others, e.g. Galois Theory, Representation Theory and Algebraic Geometry.
Three hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists