# Coursebooks

## Rings and modules

Svaldi Roberto

English

#### Summary

The students are going to solidify their knowledge of ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra.

#### Content

-basic definitions of module theory

-the fundamental theorem of finitely generated modules over a principal ideal domain

-Jordan normal form

-homological algebra

-Hilbert's nullstellensatz

-Krull dimension

-transcendence degree

-localization

-tensor product

-integral extensions

-Noether normalization

-going up theorem

-going down theorem

-primary decomposition

#### Learning Prerequisites

##### Required courses

• Linear algebra
• Théorie des groupes
• Anneaux et corps

#### Learning Outcomes

By the end of the course, the student must be able to:
• Manipulate modules over rings.
• Distinguish between properties of modules and rings
• Characterize finitely generated modules over a PID.
• Analyze rings and modules
• Apply the main theorems of the class

#### Teaching methods

ex chatedra course with exercise session

#### Assessment methods

The final grade will be assigned based on the cumulative points of the student obtained from handed in homework solutions and from the written exam. The weights of the two parts are:

25% - homework

70 % - written exam

There will be 4 homeworks that students will be required to hand in on dates to be determined at the start of the course.

Dans le cas de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés.

#### Resources

##### Notes/Handbook

There will be pdf notes provided for the course.

### In the programs

• Mathematics, 2019-2020, Bachelor semester 5
• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Rings and modules
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

MoTuWeThFr
8-9MAA110 MAA112
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German