*Instructor:* Dr. Mátyás DOMOKOS

*Prerequisite:* An introductory algebra course covering such basic concepts as groups, rings, fields and ideals of a ring.

*Text:* Miles Reid: Undergraduate Commutative Algebra

and selected topics from

M. F. Atiyah and I. G. Macdonald: Introduction to Commutative Algebra

H. Matsumura: Commutative Rings

*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.