Commutative algebra

Teacher(s)

Vitale Enrico;

Language

French

Prerequisites

Linear Algebra (LMAT1131 or equivalent).

Main themes

Rings and their ideals. In particular, local rings and polynomial rings. Modules over a ring.

Learning outcomes

At the end of this learning unit, the student is able to :
1	Establish the link between congruences and ideals. Distinguish between different types of ideals by their quotient rings and locate a ring. Deal with problems of divisibility and decomposition in Noetherian, factorial, principal and Euclidean rings. Appreciate the difference between modules and vector spaces. Recognise the role of rings and modules in other courses in the Bachelor's degree programme in mathematics.

Content

This course introduces abstract algebraic notions playing an important role throughout the cursus of Bachelor and Master in Mathematics : commutative rings and modules.
The following topics are discussed :
- Commutative rings and ideals, quotient rings, isomorphism theorems.
- Integral domains, local rings, localisations, fields of fractions.
- Maximal ideals and Krull's theorem. 
- Polynomial rings. Euclidean rings, unique factorization domains (UFD). 
- Gauss' theorem : if A is a UFD, then the polynomial ring A[X] is UFD.
- Noetherian rings, Hilbert's basis theorem.
- Modules, direct sums and direct products, free modules and projectives, modules of finite type.
- (Time permetting) Exact sequences, tensor products.

Teaching methods

Learning activities consist of lectures including supervised exercise sessions The lectures aim to introduce fundamental concepts, to explain them by showing examples and by delineating their use, to show their reciprocal connections and their connections with other courses in the programme for the Bachelor in Mathematics.
The supervised exercise sessions aim to teach how to select the appropriate method in the resolution of exercices.
The two activities are given in presential sessions.

Evaluation methods

The assessment aims to test knowledge and understanding of concepts, examples and fundamental results, the ability to build a coherent reasoning, mastery of demonstration techniques introduced during the course. The assessment consists of a final oral exam. To establish the final grade, we will take into account the oral exam and active participation in the practical work.

Online resources

The syllabus, also including the exercise statements for the practicals, is available on the course's MoodleUCLouvain site.

Bibliography

Saunders MacLane & Garrett Birkhoff, Algebra, third edition, AMS Chelsea Publishing 1988
Hymann Bass, Algebraic K-theory, W.A. Benjamin Inc. 1968

Faculty or entity

> MATH