SUBJECT

Title

Topics in ring theory

Type of instruction

lecture + practical

Level

master

Part of degree program
Credits

3+3

Recommended in

Semesters 1-4

Typically offered in

Autumn/Spring semester

Course description

Structure theory: primitive rings, Jacobson’s Density Theorem, the Jacobson radical of a ring, commutativity theorem. Central simple algebras: tensor product of algebras, the Noether–Scolem Theorem, the Double Centralizer Therem, Brauer group, crossed product. Polynomial identities: structure theorems, Kaplansky’s theorem, the Kurosh Problem, combinatorial results, quantitative theory. Noetherian rings: Goldie’s theorems and generalizations, dimension theory. Artinina rings and generalizations: Bass’s characterization of semiperfect and perfect rings, coherent rings, von Neumann regular rings, homological properties. Morita theory: Morita equivalence, Morita duality, Morita invariance. Quasi-Frobenius rings: group algebras, symmetric algebras, homological properties. Representation theory: hereditary algebras, Coxeter transformations and Coxeter functors, preprojective, regular and preinjective representations, almost split sequences, the Baruer–Thrall Conjectures, finite representation type.

The Hom and tensor functors: projective, imjective and flat modules. Derived functors: projective and injective resolutions, the construction and basic properties of the Ext and Tor functors. Exact seuqences and the Ext functor, the Yoneda composition, Ext algebras. Homological dimensions: projective, injective and global dimension, The Hilbert Syzygy Theorem, dominant dimension, finitistic dimension, the finitistic dimension conjecture. Homological methods in representation theory: almost split sequences, Auslander–Reiten quivers. Derived categories: triangulated categories, homotopy category of complexes, localization of categories, the derived category of an algebra, the Morita theory of derived categories by Rickard.

Readings
  • Anderson, F.–Fuller, K.: Rings and categories of modules, Springer, 1974, 1995
  • Auslander, M.–Reiten, I.–Smalø: Representation theory of Artin algebras, Cambridge University Press, 1995
  • Drozd, Yu. –Kirichenko, V.: Finite dimensional algebras, Springer, 1993
  • Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras, CUP, 1988
  • Herstein, I.: Noncommutative rings. MAA, 1968.
  • Rotman, J.: An introduction to homological algebra, AP, 1979