Noncommutative algebra

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
core mandatory
Group:
M2
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Noncommutative division rings. Frobenius' theorem. Wedderburn's theorem on finite divison rings.
Radical. Semisimple algebras. Wedderburn's theorem. Maschke's theorem.
Simple and semisimple modules. Density theorem. Jacobson radical.
Tensor product of algebras. Skolem-Noether theorem. Double centralizer theorem. Brauer group.

Readings

R. K. Dennis, B. Farb, Noncommutative algebra, Springer, 1993.
T. Y. Lam, A first course in noncommutative rings, Springer, 2001.
R. S. Pierce, Associative algebras, Springer, 1982.
L. Rowen, Graduate algebra: Noncommutative view, AMS, 2008.
M. Brešar, Introduction to Noncommutative Algebra, Springer, 2014

Objectives and competences

To master basic concepts and tools of noncommutative algebra.

Intended learning outcomes

Knowledge and understanding:
Understanding of basic concepts and theorems of noncommutative algebra, and their role in some other areas.
Application:
In other mathematical areas.
Reflection:
Understanding the theory on the basis of examples and applications.
Transferable skills:
Formulation and solution of problems using abstract methods.

Learning and teaching methods

Lectures, exercises, homeworks, consultations.

Assessment

Homeworks
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Matej Brešar:
BREŠAR, Matej, CHEBOTAR, M. A., MARTINDALE, Wallace S. Functional identities, (Frontiers in mathematics). Basel, Boston, Berlin: Birkhäuser, cop. 2007. XII, 272 str. ISBN 978-3-7643-7795-3. ISBN 978-3-7643-7796-0. [COBISS-SI-ID 14332505]
BREŠAR, Matej. An elementary approach to Wedderburn's structure theory. Expositiones mathematicae, ISSN 0723-0869, 2010, vol. 28, no 1, str. 79-83. [COBISS-SI-ID 15382617]
BREŠAR, Matej. An alternative approach to the structure theory of PI-rings. Expositiones mathematicae, ISSN 0723-0869, 2011, vol. 29, no 1, str. 159-164. [COBISS-SI-ID 15859545]
Jaka Cimprič:
CIMPRIČ, Jaka. Free skew fields have many [ast]-orderings. Journal of algebra, ISSN 0021-8693, 2004, vol. 280, no. 1, str. 20-28. [COBISS-SI-ID 13210201]
CIMPRIČ, Jaka. Formally real involutions on central simple algebras. Communications in algebra, ISSN 0092-7872, 2008, vol. 36, no. 1, str. 165-178. [COBISS-SI-ID 14613337]
CIMPRIČ, Jaka, HELTON, J. William, MCCULLOUGH, Scott, NELSON, Christopher. A noncommutative real nullstellensatz corresponds to a noncommutative real ideal: algorithms. Proceedings of the London Mathematical Society, ISSN 0024-6115, 2013, vol. 106, iss. 5, str. 1060-1086. [COBISS-SI-ID 16636249]