Rings are the basic mathematical entities in which we calculate. The most elementary examples are fields and the resulting theory is completely covered in the algebra courses in years 1 and 2.

In this course we study linear algebra over more general commutative rings; here the notion of ‘vector space over a field’ is replaced by that of ‘module over a ring’. In this wider context many of the familiar aspects of linear algebra require modification. For example the ‘basis theorem’ for vector spaces is replaced by the notion of ‘free resolution’ for more general modules.

We will concentrate on a class of ‘well behaved’ rings, the so-called Noetherian rings. Examples are the rings F[t1, . . . , tn], Z[t1, . . . , tn] of polynomials over a field F and integers Z. As an analogue and modification of the Jordan Normal Form Theorem we shall classify modules over Principal Ideal Domains (PIDs). A special case is the classification theorem for finitely generated abelian groups.

We shall introduce the notion of the ‘dimension of a ring’. This indicates how far from being a field a given ring is. Thus fields have dimension 0, Z has dimension 1, whilst the polynomial ring Z[t1, . . . , tn] has dimension n + 1.

This course will provide a solid foundation of commutative rings and module theory, as well as help developing foundational notions helpful in other areas such as number theory, algebraic geometry, and homological algebra.

MATHEMATICS|PURE MATHEMATICS|ANALYSIS|FUNCTIONAL ANALYSIS|BANACH SPACES|HILBERT SPACES

Elementary analysis mostly studies real-valued functions on the real line R or on n-dimensional space Rn. Functional analysis, by contrast, shifts the point of...

Elementary analysis mostly studies real-valued functions on the real line R or on n-dimensional space Rn. Functional analysis, by contrast, shifts the point of view: we collect all the functions of a given class (for instance, all bounded continuous functions) into a space of functions, and we study that space (and operations on it) as an object in its own right. Since spaces of functions are nearly always infinite-dimensional, we are led to study analysis on infinite-dimensional vector spaces, of which the most important cases are Banach spaces and Hilbert spaces. This course provides an introduction to the basic concepts of functional analysis. These concepts are crucial in the modern study of partial differential equations, Fourier analysis, quantum mechanics, probability and many other fields.