Matematikus mesterszak: Gyűrűk és algebrák (2024 ősz)
– a kurzus leírása /
MSc in Mathematics: Rings and Algebras (Fall 2024) – Course outline
The course will give a brief introduction into some important notions of
the theory of rings and associative algebras. Some of the topics discussed in
the course:
- primitive rings, prime rings
- the Jacobson radical
- commutativity theorems
- module decompositions
- chain conditions
- projective and injective modules
- categories and functors
- some generalizations of artinian rings
- Morita theory
- chain complexes and homology modules
Prerequisites:
- basic courses in linear and abstract algebra (algebra 1 to algebra 4 or
equivalent)
- basic notions like rings and algebras (left, right, two-sided) ideals,
quotient rings; modules, quotient modules, direct sums; ring and
module homomorphisms; familiarity with standard examples for rings
(like polynomial or matrix rings) and for modules (e.g. vector spaces
and abelian groups); etc.
Marking:
- The grade for the problem session (tutorial part) is based on
marked assignments. There will be 2 sets of assigment problems,
consisting of approx. 10 problems each. The first assignment sheet
will be given shortly before the fall break (middle of October), the
other one in late November. You will have at least three weeks to hand
in the solutions. 70% or more will be needed for grade 5, and 40% or
more for grade 2. Presented problem solutions during the term will be
additionally taken into account towards your term grade.
- There will be a written final exam at the end of the course. In the exam
definitions, statements and proofs will be asked.
Textbooks:
We shall not follow a single textbook, however many books will be recommended
during the first lecture. Course notes will be made available.