Commutative Algebra — CMA

• Instructor: Dr. Mátyás DOMOKOS
• Contact: domokos dot matyas at renyi dot hu
• Prerequisites: An introductory algebra course covering such basic concepts as groups, rings, fields and ideals of a ring.
• Text: Course notes provided after the lectures
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.