# COMMUTATIVE ALGEBRA

## Learning outcomes of the course unit

To define and study commutative rings in general and, in particular, Noetherian rings and Dedekind domains with emphasis on their extensions (integral or not integral), structure theorems and decomposition theorems (for ideals and modules).

## Prerequisites

A basic course in Algebra (groups, rings and fields).

## Course contents summary

Rings, ideals and modules.

Exact sequences, localization and tensor product.

Noetherian rings, Artinian rings and Dedekind domains.

Completion and dimension theory.

## Course contents

Rings ideals and modules: basic definitions, prime and maximal ideals, nilpotents, finitely generated modules, homomorphisms, exact sequences, snake lemma, extension of scalars (localization and tensor product).

Noetherian rings, Artinian rings, Dedekind domains: definitions and basic properties, Hilbert basis Theorem, integral extensions, going up and going down theorems, decomposition of ideals as intersections of irreducibles, local Dedekind domains, unique factorization of ideals in Dedekind domains.

Completion: topologies, inverse and direct limits, I-adic completion, filtrations.

Dimension theory: equivalent definitions of dimension, dimension of local noetherian rings.

Whenever possible we shall point out the deep relation between the results of commutative algebra presented here and the corresponding theorems in algebraic geometry.

## Recommended readings

M.F. Atiyah - I.G. Macdonald "Introduction to commutative algebra"

S. Lang "Algebra"

D Eisenbud "Commutative Algebra: with a View Toward Algebraic Geometry"

## Teaching methods

The preferred teaching tool for the development of such knowledge are the lectures.

Taking notes is seen as part of the learning process.

A large part of the topics shall be presented and/or developed using the numerous exercises that can be found in the textbooks.

## Assessment methods and criteria

The assessment of learning is done in classic way, through the evaluation of an oral interview. In the colloquium, the student must be able to independently conduct demonstrations and solve exercises using an appropriate algebraic language and a proper mathematical formalism.