06689 - Commutative Algebra

Course Unit Page

Academic Year 2022/2023

Learning outcomes

At the end of the course the student will have acquired the fundamentals of the theory of finitely generated modules, will be able to recognize injective and projective modules and will have learned skills about the localization of rings and modules. He will be able to apply the knowledge acquired for solving problems and for producing proofs.

Course contents

Unital commutative rings, ideals, homomorphisms, quotient rings. Integrity domains, zero dividers, nilpotent elements, fields. Prime ideals and maximal ideals. Local rings and their characterization. Operations on ideals.

Modules, submodules and their operations. Exact sequences of modules, the snake lemma. Projective and injective modules. Finitely generated modules, free modules. Nakayama's lemma.

Rings of fractions and localizations. Primary decomposition of ideals.

Integral elements, integral extensions of rings and integral closure.

Conditions on chains. Artinian and Noetherian rings and modules. Hilbert's basis theorem. Normalization lemma and Nullstellensatz.

Dedekind domains.


M.F.Atiyah - I.G.Macdonald: Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Reading Massachusetts, 1969.

D.G.Northcott: Ideal Theory Cambridge University Press, Cambridge 1953

D.Cox - J.Little - D.O'Shea: Ideals, Varieties and Algorithms. 4th Ed. Undergraduate Texts in Mathematics. Springer Verlag, New York 2007

Altman, Allen & Kleiman, Steven. (2013). A term of Commutative Algebra (https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf.)

Teaching methods

Lectures and exercise sessions; team works. Sheets of exercises will be handed out during the lectures, in addition to the ones available in the suggested textbooks.

In the office hours students will be coached individually.

Assessment methods

Oral exam, starting from the discussion of the exercises that students must solve and give to the teachers at least a week before the exam. For the solution of some exercises students need to use COCOA or Singular or some other software for symbolic computation available in the Laboratories.

The date of the exam must be fixed with the teachers of the course.

Teaching tools

Blackboard, photocopies, projector. Symbolic algebraic calculation software.

Office hours

See the website of Roberto Pagaria

See the website of Mirella Manaresi