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Description
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 鈥榲ector space over a field鈥� is replaced by that of 鈥榤odule over a ring鈥�. In this wider context many of the familiar aspects of linear algebra require modification. For example the 鈥榖asis theorem鈥� for vector spaces is replaced by the notion of 鈥榝ree resolution鈥� for more general modules.
We will concentrate on a class of 鈥榳ell behaved鈥� 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 鈥榙imension of a ring鈥�. 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.
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 8th April 2024.
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