Instructor: Dr. József PELIKÁN
Text: M. F. Atiyah -- I. G. MacDonald: Introduction to
Commutative Algebra and/or Miles Reid: Undergraduate Commutative Algebra
The "official" textbook of the course is the Atiyah-MacDonald book, which can also be found here on-line (note that you need to wait a bit and then choose download at the bottom of the page)
Prerequisite: An introductory algebra course covering such basic concepts as groups, rings, fields and ideals of a ring.
'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.
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