- Lecturer: Damian Rossler

General Prerequisites:

Rings and Modules is essential. Galois Theory is strongly recommended.

Course Term: Hilary

Course Lecture Information: 16 lectures

Course Weight: 1

Course Level: H

Assessment Type: Written Examination

Course 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.

Course Synopsis:

Modules, ideals, prime ideals, maximal ideals.

Noetherian rings; Hilbert basis theorem. Minimal primes.

Localization.

Polynomial rings and algebraic sets. Weak Nullstellensatz.

Nilradical and Jacobson radical; strong Nullstellensatz.

Integral extensions. Prime ideals in integral extensions.

Noether Normalization Lemma.

Krull dimension; dimension of an affine algebra.

Noetherian rings of small dimension, Dedekind domains.

Noetherian rings; Hilbert basis theorem. Minimal primes.

Localization.

Polynomial rings and algebraic sets. Weak Nullstellensatz.

Nilradical and Jacobson radical; strong Nullstellensatz.

Integral extensions. Prime ideals in integral extensions.

Noether Normalization Lemma.

Krull dimension; dimension of an affine algebra.

Noetherian rings of small dimension, Dedekind domains.