# SUPERIOR ALGEBRA

## Learning outcomes of the course unit

Knowledge and understanding

The student will be introduced to the basic concepts and techniques of commutaive algebra and algebraic geometry.

Applying knowledge and understanding

The student will be able to: i) solve simple exercise of algebraic geometry, ii) simple

exercises of commutaive algebra.

## Prerequisites

Theory of groups.

## Course contents summary

The course is an introduction to the basic notions of commutative algebra and basic notions of algebraic geometry.

The first part studies commutative rings with unit ideals, Nullstellensatz Theorem, Zarisky Topology while the second part is devoted to the study of modules, operations on modules, Hamilton-Cayley Theorem, Nakayama Lemma and flat modules.

In the third part of the course we study localization of rings and modules, primary decomposition, Noetherian and Artin rings (modules) , Hilbert basis Theorem .

The course ends with a study of integral dependence and valutations, Krull dimension and basic notions on algebraic varieties.

## Course contents

Commutative rings with unit, prime ideal, radical, Nilradical, Nullestellensatz Theorem, Zarisky Topologyt.

Modules, operations on modules, Hamilton-Cayley Theorem and Lemma di Nakayama.

Localizations of rings and modules, primary decomposition, Noetherian and Artin rings, Hilbert basis Theorem.

Integral dependence and valutations, Krull dimension and basic notions on algebraic varieties.

## Recommended readings

Atyah e Mc Donald,

Algebra commutativa

## Teaching methods

During lectures, the material of the course is presented using formal

definitions and proofs; abstract concepts are illustrated through

significant examples, applications, and exercises. The discussion of

examples and exercises is of fundamental importance for grasping the

meaning of the abstract mathematical concepts.

## Assessment methods and criteria

Course grades will be based on an oral interview.

In the colloquium, students should establish that students have learned the course

materials to a sufficient level and should be able

to repeat definitions, theorems and proofs given in the lectures using a

proper mathematical language and formalism. In the colloquium, students should be able

to repeat definitions, theorems and proofs given in the lectures using a

proper mathematical language and formalism.