Teacher(s)
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 : | |
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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.
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.
In order to test the new academic rhythms (bimonthly), the course will be given during the first six weeks of Q1, with 4 hours of lectures and 2 hours of tutorials each week.
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.
In order to test the new academic rhythms (bimonthly), the course will be given during the first six weeks of Q1, with 4 hours of lectures and 2 hours of tutorials each week.
Evaluation methods
The evaluation aims to test knowledge and understanding of fundamental concepts, examples and results, the ability to construct coherent reasoning, and mastery of the demonstration techniques introduced during the course.
For the final evaluation, each student will be able to choose between an oral exam and a written exam. The oral exam will take place by appointment in week 7 and consists of a presentation of approximately half an hour on an argument chosen by the student followed by questions which may relate to the entire subject. The one-hour written exam (without access to external resources) consists of a tutorial-type exercise and a more theoretical question. The written exam will take place either at the end of week 7 or at the beginning of week 8.
The student remains responsible for the content of their productions (both oral and written), regardless of the sources used (AI in particular).
To establish the final grade, the final exam and active participation in practical work will be taken into account.
There will be no assessment in January, but there will be the possibility of taking the resit exam in August, under the same conditions described above.
For the final evaluation, each student will be able to choose between an oral exam and a written exam. The oral exam will take place by appointment in week 7 and consists of a presentation of approximately half an hour on an argument chosen by the student followed by questions which may relate to the entire subject. The one-hour written exam (without access to external resources) consists of a tutorial-type exercise and a more theoretical question. The written exam will take place either at the end of week 7 or at the beginning of week 8.
The student remains responsible for the content of their productions (both oral and written), regardless of the sources used (AI in particular).
To establish the final grade, the final exam and active participation in practical work will be taken into account.
There will be no assessment in January, but there will be the possibility of taking the resit exam in August, under the same conditions described above.
Other information
Any use of artificial intelligence within the framework of this course must comply with the guidelines established by the Faculty of Science.
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
Hymann Bass, Algebraic K-theory, W.A. Benjamin Inc. 1968
Faculty or entity
