6 ECTS credits
150 u studietijd

Aanbieding 1 met studiegidsnummer 4013373FNR voor alle studenten in het 1e semester met een gespecialiseerd master niveau.

Semester
1e semester
Inschrijving onder examencontract
Niet mogelijk
Beoordelingsvoet
Beoordeling (0 tot 20)
2e zittijd mogelijk
Ja
Onderwijstaal
Engels
Onder samenwerkingsakkoord
Onder uitwisselingsakkoord mbt studiedelen
Faculteit
Faculteit Wetenschappen en Bio-ingenieurswetensch.
Verantwoordelijke vakgroep
Wiskunde
Onderwijsteam
Claudio Leandro Vendramin (titularis)
Onderdelen en contacturen
30 contacturen Hoorcollege
30 contacturen Werkcolleges, practica en oefeningen
Inhoud

The aim is to get a good insight in the algebraic structure of important classes of modules and rings. A very solid knowledge is expected of all notions and proves. Via independent work one must be able to understand and prove related properties. 

The reference book for this course is  ``Lam, T. Y. A first course in non-commutative rings, GTM 131''.
This book contains many excercises and it is expected that the student solves many of these. The solutions must be communicated both written and oral.

Chapter 1:  Modules and Semisimple Rings

  • Noetherian  and Artinian modules, Projective and Injective Modules, Sesimple Modules and Rings , Wedderburn-Artin Theory

Chapter 2: The Jacobson Radical

  • The Jacobson radical, Nil and nilpotent ideals, Semiprime Rings, Semiprimitive Rings, Wedderburn rings, Hopkins, Kevitzky and Nakayama results, Von Neumann Regular rings, Linear groups and the Burnside problem

Chapter 3: Prime and Primitive Rings

  • Prime rings, primitive rings, density theorem

Chapter 4: Skew Fields

  • Wedderburn's Theorem, Additive and multiplicative commutators, commutativity theorems, algerbaic skew fields

Chapter 5: Goldie theorems

Studiemateriaal
Digitaal cursusmateriaal (Vereist) : Cursusnota's beschikbaar op http://homepages.vub.ac.be/~efjesper, http://homepages.vub.ac.be/~efjesper
Handboek (Aanbevolen) : Lectures on modules and rings. Graduate Texts in Mathematics, Lam, T. Y., Springer-Verlag, New York, 9781461268024, 1999
Handboek (Aanbevolen) : A first course in noncommutative rings, Lam, T. Y., 2de, Springer-Verlag, New York, 9780387953250, 2001
Bijkomende info

Course notes



Complementary study material: Lam, T. Y. A first
course in noncommutative rings. Second edition.
Graduate Texts in Mathematics, 131.
Springer-Verlag, New York, 2001. xx+385 pp. ISBN:
0-387-95183-0 16-01

Lam, T. Y. Lectures on modules and rings.
Graduate Texts in Mathematics, 189.
Springer-Verlag,

Leerresultaten

Algemene competenties

1. Student knows and has insight in the fundamental results of important classes of rings.
2. Student can look up related properties and structures.
3. Student can prove related properties.
4. Student can make connections with realted  concepts and other theories.
5. Student can think in function of problem.
6. Student can synthesize and interpret results.
7. Student independently can look up and solve  exercises.
8. Student can analyze results.
9. Student can consult and understand recent literature.
10. Student can independently compose a correct mathematics text about the solutions of exercises.
11. Student can draw up a text on another theory independently and report orally.

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

Examination:
Theoretical part 70%: oral exam. Evaluated are both the knowledge of all notions and proofs and the global  picture of the material. 

Excercise part  30%: at the beginning of the oral exam, the student hands in a document with the solutions of 30 excercises (this is a choice of the most challenging excercises the student has solved). During the first part of the oral exam, the student gives a short presentation of the three most challenging problems that have been solved.
The examination mark on this part takes into account the difficulty level of the chosen excercises and the originality and correctness of the solutions.

 

A mark will only be asigned if the student particpates in all exams, tests and assignments.

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:
Master in de wiskunde: fundamentele wiskunde
Master in de wiskunde: onderwijs
Educatieve master in de wetenschappen en technologie: wiskunde (120 ECTS, Etterbeek)