General Prerequisites: Rings and Modules is essential. Galois Theory is strongly recommended.
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.
