Instructor: Dr. István ÁGOSTON
Anderson, F..Fuller, K.: Rings and categories of modules, Springer, 1974, 1995
Herstein, I.: Noncommutative rings. MAA, 1968.
Lam, T.Y.: A first course in non-commutative rings, Springer, 1991
Rotman: An introduction to homological algebra, Academic Press 1979
Associative rings and algebras. Constructions: polynomials, formal power series, linear operators, group algebras, path algebras. free algebras, tensor algebras, exterior algebras. Structure theory. Primitive rings, density theorem, the Jacobson radical, commutativity theorems. Direct sum decomposition of modules, Azumaya.s theorem. Chain conditions, injective modules. Theorems of Hopkins and Levitzki.
Categories and functors. Algebraic and topological examples. Natural transformations. The concept of categorical equivalence. Covariant and contravariant functors. Properties of the Hom and tensor functors (for non-commutative rings). Adjoint functors. Additive categories, exact functors. The exactness of certain functors: projective, injective and flat modules. Morita theory for rings.
Generalizations of artinian rings: semiperfect and perfect rings. Direct decomposition of projective modules.
Homological algebra. Chain complexes, homology groups, chain homotopy. Examples from algebra and topology. The long exact sequence of homologies.