6 ECTS credits
150 h study time

Offer 1 with catalog number 4013373FNR for all students in the 1st semester at a (F) Master - specialised level.

Semester
1st semester
Enrollment based on exam contract
Impossible
Grading method
Grading (scale from 0 to 20)
Can retake in second session
Yes
Taught in
English
Partnership Agreement
Under agreement for exchange of courses
Faculty
Faculty of Sciences and Bioengineering Sciences
Department
Mathematics
Educational team
Claudio Leandro Vendramin (course titular)
Activities and contact hours

30 contact hours Lecture
30 contact hours Seminar, Exercises or Practicals
Course Content

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

Course material
Digital course material (Required) : Cursusnota's beschikbaar op http://homepages.vub.ac.be/~efjesper, http://homepages.vub.ac.be/~efjesper
Handbook (Recommended) : Lectures on modules and rings. Graduate Texts in Mathematics, Lam, T. Y., Springer-Verlag, New York, 9781461268024, 1999
Handbook (Recommended) : A first course in noncommutative rings, Lam, T. Y., 2de, Springer-Verlag, New York, 9780387953250, 2001
Additional 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,

Learning Outcomes

General competencies

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.

Grading

The final grade is composed based on the following categories:
Other Exam determines 100% of the final mark.

Within the Other Exam category, the following assignments need to be completed:

  • examen with a relative weight of 100 which comprises 100% of the final mark.

Additional info regarding evaluation

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.

Allowed unsatisfactory mark
The supplementary Teaching and Examination Regulations of your faculty stipulate whether an allowed unsatisfactory mark for this programme unit is permitted.

Academic context

This offer is part of the following study plans:
Master of Mathematics: Fundamental Mathematics (only offered in Dutch)
Master of Mathematics: Education (only offered in Dutch)
Master of Teaching in Science and Technology: wiskunde (120 ECTS, Etterbeek) (only offered in Dutch)