Course content

The contents of the course may vary, but it will have a kernel consisting of ideals, modules, chain conditions, the spectrum of a ring, Hilberts Nullestellensatz, associated primes and decomposition theorems, integral elements and rings, valuation rings, Dedekind rings, graded rings, dimension theory. It can also include regular sequences, Koszul complex, and finally regular, Cohen-Macaulay and Gorenstein rings.

Learning outcome

1. Knowledge. The contents of the course may vary, but it will have a core consisting of the following: Ideals, prime and maximal ideals, Hilbert's Nullstellensatz, modules, operations on modules, localizations with respect to multiplicatively closed sets and with respect to prime ideals, local rings and local properties, localization of modules, chain conditions, noetherian and artinian rings, dimension theory. It may moreover include Dedekind domains, discrete valuation rings, completions, regular sequences, Koszul complexes, Cohen-Macaulay rings, Gorenstein rings.

2. Skills. The students should learn the topics mentioned above and be able to conduct researches in commutative algebra and its applications.

3. Competence. The students should be able to participate in scientific discussions and begin with own reseach in commutative algebra.

Learning methods and activities

Lectures, possibly as a guided self-study.

The course is taught every third year, if there are enough students. Next time spring 2026. If there are few students, there will be guided self-study.

Further on evaluation

Oral exam.

Recommended previous knowledge

MA3201 Rings and modules, MA3204 Homological algebra. Having also taken MA3202 Galois theory and MA3203 Ring theory will be an advantage.

Course materials

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, together with lecture notes.