*Instructor: * Dr. József PELIKÁN

*Text:* M. F. Atiyah -- I. G. MacDonald: Introduction to
Commutative Algebra and/or Miles Reid: Undergraduate Commutative Algebra

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

*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.

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

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

**Noetherian rings.** Chain conditions for modules and rings.
Hilbert basis theorem. Primary ideals. Primary decomposition, the
Lasker-Noether theorem. Krull dimension. Artinian rings.

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

**Integral dependence.** Integral closure. The going-up and
going-down theorems. Valuations. Discrete valuation rings. Dedekind
domains. Fractional ideals.

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

**Dimension theory. **Various dimensions. Krull's principal ideal
theorem. Hilbert functions. Regular local rings. Hilbert's syzygy
theorem.