6 ECTS credits
150 u studietijd

Aanbieding 1 met studiegidsnummer 1010311ANR voor alle studenten in het 1e semester met een inleidend bachelor niveau.

Semester
1e semester
Inschrijving onder examencontract
Niet mogelijk
Beoordelingsvoet
Beoordeling (0 tot 20)
2e zittijd mogelijk
Ja
Inschrijvingsvereisten
Students must have followed ‘Introduction to Group Theory’, before they can enroll for ‘Ring and Module Theory’.
Onderwijstaal
Engels
Faculteit
Faculteit Wetenschappen en Bio-ingenieurswetensch.
Verantwoordelijke vakgroep
Wiskunde
Onderwijsteam
Geoffrey Janssens
Claudio Leandro Vendramin (titularis)
Onderdelen en contacturen
26 contacturen Hoorcollege
26 contacturen Werkvormen en Praktische Oef.
Inhoud

We give an introduction in the algebraic structure of rings, ideals, modules and group representations. The core of the course is to deduce important conclusions from elementary assumptions. At the same time we try to show the ``art'' in the science ``algebra''.

 

The emphasis is on getting familiarised with abstract strucutres and on understanding the results and their proofs. Furthermore, one has to learn to prove indepently related results. The tutorials will be very crucial for the latter.

 

In order to acquire the necessary mathematical skills and to improve their communication skills, students will have to make two projects (homeworks). Also an introduction to the programming language TEX will be given: this is a program that allows to write mathematical texts. For the homework  every student will receive some tasks: these can be excercises discussed during the tutorials as well as completely new excercises. It is expected that these are solved in full detail and correctly and that they are  presented in a document made using TEX. Solutions have to be submitted by email and also in a hard copy version. The project will have to be defended during an oral presentation. It is expected that in a short presentation (maximum 15 minutes) the student will give a clear  outline for the other students. The evaluation will be based on the mathematical content, the actual presentation and the use of TEX.

 

The fourth chapter contain some applications and extensions of the first three chapters. Depending on the available time, this chapter could be treated partially or completely.

 

Course Content:

 

Chapter 1: Representation Theory

1.1 Definition of representations

1.2 Examples of representations

1.3 Subrepresentations

1.4 Irreducible representations

1.5 Tensor product of representations

1.6 Characters of representations

1.7 The Lemma of Schur

1.8 Othogonality relations

1.9. The regular reperesentation

1.10 Class functions

1.11 Examples

 

Chapter 2: Rings and ideals

2.1 Definitions and  Examples

2.2 Subrings

2.3 Ring homomorphisms

2.4 Ideals

2.5 Isomorphism theorems for rings

2.6 Prime ideals en maximal ideals

2.7 Maximal ideals

2.8 Noetherian rings

2.9 The Chinese remainder theorem

2.10 Fields of fractions

2.11 Principal and Euclidean rings

2.12 Unique factorsiation domains

 

Chapter 3: Modules

3.1 Modules and submodules

3.2 Homomorphism and quotient modules

3.3 Modules and group representations

3.4 Free Modules

3.5 Finitely generated modules

3.6 Modules and Euclidean domains

3.7 Exact rows and projective modules

 

Chapter 4: Applications

4.1 Fields and field extensions

4.2 The theorem of Cayley-Hamilton

4.3 The field of p-adic numbers

4.4 Lie algebras and Lie groups

 

Chapter 5: Exercises

Studiemateriaal
Digitaal cursusmateriaal (Vereist) : Cursus, http://homepages.vub.ac.be/~efjesper/
Handboek (Aanbevolen) : Algebra, Groups, Rings and Fields, L. Rowen, A.K. Peters, Welleslley, 9780367449230, 2019
Handboek (Aanbevolen) : Algebra, Vol. 1, P.M. Cohn, BIB, 9780471164319, 1974
Handboek (Aanbevolen) : Algebra, M. Artin, 2de, Pearson, 9781292027661, 2014
Bijkomende info

Course notes are available on:
http://homepages.vub.ac.be/~efjesper/



Complementary study material:
Complementary study material
M. Artin, Algebra, Prentice Hall, London,
1991. (ISBN: 0-13-004763-5).

P.M. Cohn, Algebra, Vol. 1, John Wiley \& Sons,
London, 1974. (ISBN: 0-471-16431-3)

L. Rowen, Algebra, Groups, Rings and Fields,
A.K. Peters, Welleslley, 1994. (ISBN: 1-56881-028-8)

Leerresultaten

Algemene competenties

1. Student knows basic concepts from representation theory of groups, ring- and module theory.
2. Student is able to apply theoretical concepts from the theory on examples.
3. Student can think in function of problem solving, both individually and in group work.
4. Student can reconstruct proofs.
5. Student can prove independently related results.
6. Student can make connections with other theories, including group theory and functional analysis.
7. Student can write a mathematical text independently about solutions of exercises.
8. Student knows the sofwarepackage LaTex and uses it for communicating clearly presented solutions of excercises.
9. Student can orally clearly present the solutions of excercises.
10. Student can consult standard references. 

Beoordelingsinformatie

De beoordeling bestaat uit volgende opdrachtcategorieën:
Examen Andere bepaalt 100% van het eindcijfer

Binnen de categorie Examen Andere dient men volgende opdrachten af te werken:

  • examen met een wegingsfactor 100 en aldus 100% van het totale eindcijfer.

Aanvullende info mbt evaluatie

The evaluation consists of two parts, each contributing 50% to the final score.

Part 1: Theoretical examination during the examination period: written preparation with written discussion of the questions.

Part 2: consisting of problems and exercises: this part consists of a written examination during the examination period (contributing 30% to the final score) and the results of written tasks/projects during the term (contributing 20% to the final score). The score of part 2 is determined by the written examen and the results of the one or more tasks performed during the term.

To succeed for this course, the average score of parts 1 and 2 must be at least 10/20. Furthermore, the score on each of the parts must be at least 7/10. If the score of one of the parts is lower than 7/20, the final score will be the average of both parts if this is less than 7/20 or 7/20 if the average is higher than 7/20.

A final score will only be awarded if the student has participated in all parts of the complete evaluation. However, a score of 10/20 or higher for a particular part of the evaluation will result in a dispensation of this part during the second try examination period and during the examination periods of the next academic year. The student can decide not to accept this dispensation. In such case, the student must mention this non-acceptance by e-mail to the responsible lecturer of this course, not later than August 15th (in case of non-acceptance for the second try examination), or not later than November 1st (in case of non-acceptance for the next academic year). Note that the non-acceptance of a dispensation is irrevocable.

Toegestane onvoldoende
Kijk in het aanvullend OER van je faculteit na of een toegestane onvoldoende mogelijk is voor dit opleidingsonderdeel.

Academische context

Deze aanbieding maakt deel uit van de volgende studieplannen:
Bachelor in de wiskunde en Data Science: Standaard traject