Synopsis for C2.2a: Commutative Algebra

Level: M-level \hfill Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code tbc).

Recommended Prerequisites

A thorough knowledge of the second-year algebra courses, in particular rings, ideals and fields.

Overview

Amongst the most familiar objects in mathematics are the ring of integers and the polynomial rings over fields. These play a fundamental role in number theory and in algebraic geometry, respectively. The course explores the basic properties of such rings, and introduces the key concept of a module, which generalizes both abelian groups and the idea of a linear transformation on a vector space.

Synopsis

Introduction to modules. The structure of modules over a principal ideal ring. Prime ideals, maximal ideals, nilradical and Jacobson radical. Noetherian rings; Hilbert basis theorem. Minimal primes. Artin-Rees Lemma; Krull intersection theorem. Integral extensions. Prime ideals in integral extensions. Noether Normalization Lemma. Hilbert Nullstellensatz, maximal ideals. Krull dimension; Principal ideal theorem.; dimension of an affine algebra.

1. Atiyah, Macdonald, Introduction to Commutative Algebra, (Addison-Wesley, 1969).
2. Eisenbud Commutative Algebra: with a View Toward Algebraic Geometry, Grad. Texts Math. 150, (Springer-Verlag, 1995) Chapters 4, 5 and 13.