Course content

This course gives an introduction to commutative rings and their modules. We study concepts such as localization, decomposition of modules, chain conditions for rings and modules, and dimension theory. The course gives a relevant background for studies in algebraic geometry, but also relates the theory to problems in number theory.

Learning outcome

After completing the course you

  • know the definition of commutative rings, local rings, prime and maximal ideals, and modules over commutative rings
  • are familiar with the notions of noetherian and artinian rings and modules
  • know how to localize rings and modules, and are familiar with important applications of localization
  • know the Hilbert basis theorem and the Hilbert Nullstellensatz
  • are familiar with the concepts of support and associated primes
  • know the definition of an exact sequence of modules, and you also know important properties and applications of exact sequences
  • know the concept of direct limit and you can compute this limit in some non-trivial examples
  • know how to define tensor products of modules and are familiar with the concept of flatness
  • know Krull-Cohen-Seidenberg theory
  • know the basic results in the dimension theory for local rings
  • know how to complete a ring in an ideal.

Admission to the course

Students admitted at UiO must apply for courses in Studentweb. Students enrolled in other Master's Degree Programmes can, on application, be admitted to the course if this is cleared by their own study programme.

Nordic citizens and applicants residing in the Nordic countries may apply to take this course as a single course student.

If you are not already enrolled as a student at UiO, please see our information about admission requirements and procedures for international applicants.

Overlapping courses


4 hours of lectures/exercises per week throughout the semester.

The course may be taught in Norwegian if the lecturer and all students at the first lecture agree to it.


Final written exam or final oral exam, which counts 100 % towards the final grade.

The form of examination will be announced by the lecturer by 15 October/15 March for the autumn semester and the spring semester respectively.

This course has 1 mandatory assignment that must be approved before you can sit the final exam.

Examination support material

No examination support material is allowed.

Language of examination

Courses taught in English will only offer the exam paper in English. You may submit your response in Norwegian, Swedish, Danish or English.

Grading scale

Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. Read more about the grading system.

Resit an examination

This course offers both postponed and resit of examination. Read more:

Special examination arrangements, use of sources, explanations and appeals

See more about examinations at UiO

Last updated from FS (Common Student System) May 25, 2022 12:26:55 PM

Facts about this course

Teaching language