# B2.2 Commutative Algebra - Material for the year 2020-2021

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Rings and Modules is essential. Representation Theory and Galois Theory are recommended.

16 lectures

### Assessment type:

- Written Examination

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.

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.

- M. F. Atiyah and I. G. MacDonald:
*Introduction to Commutative Algebra*, (Addison-Wesley, 1969). - D. Eisenbud:
*Commutative Algebra with a view towards Algebraic Geometry*, (Springer GTM, 1995).