MAT.512UB / MAT.513UB Noncommutative Algebra
WS 2024/25
Lecture and exercise class.
Lecture Meeting times:
11:45-13:15 Tue @ SR11.32, Oct 1 - Jan 28
10:00-10:45 Thu @ SR11.32, Oct 3 - Jan 30 Starting time changed Oct 24!
Exercises Meeting times:
10:45-11:30 Thu @ SR11.32, Oct 3 - Jan 30 Starting time changed Oct 24!
Instructor: Eleonore Faber
Email: eleonore.faber@uni-graz.at
Office: 524 (4th floor of Heinrichstr. 36)
Office hours: by appointment
Course information (syllabus) can be found HERE.
(Handwritten) lecture notes will be posted below:
Notes week 1 : Notation, basic definitions, in particular ACC and DCC and noetherian and artinian modules.
Notes week 2 : simple and semi-simple modules, Schur's lemma, composition series.
Notes week 3 : Jordan-Hölder theorem, Krull-Schmidt property for artinian modules, a semi-simple ring is left artinian, idempotents.
Notes week 4 : more idempotents, direct sum decompositions, Jacobson density theorem.
Notes week 5 : Artin-Wedderburn theorem, Jacobson radical, properties of Jacobson radical.
Notes week 6 : Nakayama lemma, characterization of simple and semi-simple rings using the Jacobson radical, algebras.
Notes week 7 : Example: Modules over path algebras of quivers, quiver representations.
Notes week 8 : More path algebras of quivers, Gabriel's Theorem (every finite dimensional basic algebra is isomorphic to a bound quiver algebra).
Notes week 9 : Last part of proof of Gabriel's Theorem, Examples of construction of quiver of a finite dimensional algebra, prime ideals in noncommutative rings.
Notes week 10 : Semiprime rings and ideals. Student presentations.
Presentations:
Markus M. R. Tripp: Group Algebras .
Topics for the presentations:
Examples of noncommutative algebras