*Note that this course is cross-listed with the ELTE University and is held on their campus. Its schedule is set by ELTE. Time and location will be available by the beginning of September*

*Instructor:* Dr. István ÁGOSTON
*Textbooks:* none
*Suggested reading:*

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

*Course description:*

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.