Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Algebra and Number Theory

Code

gyural1u0um17em

Title

Rings and algebras (l)

Usual semester

Autumn

Published semester

2026/27/1

ECTS

3

Language

en

Learning outcomes

Knowledge: Knowledge of basic concepts, results and methods in the field. Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyse mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

Asociative rings and algebras. Constructions: polynomials, formal power series, linear operators, group algebras, free algebras, tensor algebras, exterior algebras. Structure theory: primitive rings, the density theorem, the Jacobson radical, theorems of Snapper, Amitsur, Hopkins and Levitzki, commutativity theorems. Direct sum decompositions of modules, uniqueness, theorem of Azumaya. Local rings. Injective modules, injective envelope. Chain conditions, characterization of Noetherian and Artinian rings via dierct sums of injective modules. Generalizations of Artinian rings: semiperfect and perfect rings. Direct sum decompositions of projective modules, projective cover. 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. Pre-additive categories, exactness of  functors: projective, injective and flat modules. Morita theory. Optional: Homological algebra. Chain complexes, homologies, chain homotopy. Topological and algebraic examples. The long exact sequence of homologies. Necessary prior knowledge: familiarity with basic algebraic structures (e.g. linear algebra, non-commutative rings, modules, Abelian groups, Wedderburn–Artin theorem)

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

Anderson, F.–Fuller, K.: Rings and categories of modules, Springer, 1974, 1995 Assem, I.–Simson, D.–Skowroński, A.: Elements of the representation theory of associative algebras, CUP, 2007 Herstein, I.: Noncommutative rings. MAA, 1968. Lam, T.Y.: A first course in non-commutative rings, Springer, 1991 Lam, T.Y.: Lectures on modules and rings, Springer, 1999

Programmes of the course