Short abstract:

The course presents the classical part of the theory of finite-dimensional algebras. First, are the main examples of associative algebras and the basic concepts. Considered finite-dimensional division algebra, a theorem of Frobenius.  Further sets out the elements of field theory and the theory of modules. The structure theory of finite-dimensional associative algebras is based on the concept of the Jacobson radical. We study the basic properties of the Jacobson radical, and gives examples of its calculation. In the final part of the course discusses the Wedderburn - Artin  - Molin theorem and many of its applications.

Requirements for the level of training of students who completed the study discipline "finite-dimensiona algebras"

Students who have completed the study of this discipline must:

- Understand the basic ideas and methods underlying the theory of finite-dimensional associative algebras, the role of these methods in modern mathematics and other sciences, its applications and capabilities;

- Have a theoretical knowledge of the fundamentals of the theory;

- Be able to prove the assertion of the theory.

Content of discipline.

1. Basic concepts of ring theory: subrings, ideals, quotient, product, matrix and polynomial rings, factorisation in integral (euclidean, principal ideal) domains. Finite-dimensional algebras, algebras of small dimension, Frobenius theorem.
2. Basic concepts of fields theory: field extension, simple field extensions, simple algebraic field extension. The degree of an extension, finite extensions, the Tower Theorem. Algebraic extensions, algebraic closure, the algebraic numbers, splitting fields, normal extensions. Galois Theory: automorphism groups and fixed fields,  the Galois correspondence, main Theorem of Galois Theory.
3. Basic concepts of module theory: submodules, quotient modules, direct sums, homomorphisms, finitely generated, cyclic, free and semisimple modules, modules of finite length, Artinian and Noetherian modules, matrix representation of the endomorphism rings of modules.
4. The structure theory of finite-dimensional associative algebras. Jacobson radical.
5. Wedderburn - Artin  - Molin Theorem and its applications. Maschke's theorem.

Bibliography

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in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992..

2. Beachy, J. A., Introductory Lectures on Rings and Modules, London Mathematical Society

Student Texts, Vol. 47, Cambridge University Press, Cambridge, 1999.

3. Jacobson, N., Structure of Rings, Colloquium Publications, Vol. 37, American Mathematical

Society, Providence, R.I., 1964.

4. Lam, T. Y., A First Course in Noncommutative Rings, Graduate Texts in Mathematics,

Vol. 131, Springer-Verlag, New York, 1991.

5. Lam, T. Y., Exercises in Classical Ring Theory, Problem Books in Mathematics,

Springer-Verlag, New York, 1995.

6. Lam, T. Y., Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol. 189,

Springer-Verlag, New York, 1999.

7. Lambek, J., Lectures on Rings and Modules, 3rd Ed., Chelsea Publishing Co., New

York, 1986.

8. Passman, D. S., A Course in Ring Theory, Brooks/Cole Publishing Co., Pacific Grove,

Cal., 1991.