The aim of these lectures is to introduce study of algebraic varieties. Algebraic varieties are algebro-geometric objects very useful in solving many mathematical problems.

The main parts of the course are:

1. Noetherian rings. The theorem of a basis of Hilbert. Primary decompostion of ideals in a noetherian ring.
2. Affine varieties; morphismes of affine varieties. The structure sheaf of an affine variety.
3. Projective varieties; morphismes of projective varieties. The structure sheaf of a projective variety. Connections with the theory of graduate rings.
4. Dimension. The theorem of principal ideal.
5. Birational classification of varieties.
6. Smooth and singular points.
7. The geometric model of a function field of dimension 1.
8. Hilbert-Samuel polynomial of a projective variety.