Course description: 'Commutative algebra' is the nickname of the study of commutative rings. Besides being a fascinating topic by itself, it is an indispensable tool for studying such mainstream areas of present-day mathematics as algebraic geometry and algebraic number theory. It also has applications in combinatorics and is a source of motivation for noncommutative ring theory. Topics:

Prime and maximal ideals. Prime and maximal ideals. Zorn's lemma. Nilradical, Jacobson radical. The prime spectrum.

Unique factorization domains. Gauss lemma, polynomial rings.

Modules. Operations on submodules. Finitely generated modules. Nakayama's lemma. Exact sequences. Tensor product of modules.

Noetherian rings and modules. Chain conditions for modules and rings. Hilbert basis theorem. Noether normalization lemma.

Varieties. Weak Nullstellensatz. Hilbert Nullstellensatz. Zariski topology. Coordinate ring. Singular and non-singular points. Tangent space.

Localization. Rings and modules of fractions. Extended and contracted ideals. Local properties.

Associated primes. Primary ideals. Primary decomposition, the Lasker-Noether theorem.

Integral extensions. Normality, integral closure. The going-up and going-down theorems.

Valuations. Discrete valuation rings.

Krull dimension. Transcendence degree.

Artinian rings. Finite length modules.

Dedekind domains. Class group. Fractional ideals.

Graded rings. Graded modules. Hilbert series. Hilbert-Serre theorem.

Dimension theory. Various dimensions. Krull's principal ideal theorem. Hilbert polynomial, Samuel function. Systems of parameters. Associated graded rings.