This course is an introduction to basic concepts of commutative algebra that are useful for algebraic number theory and algebraic geometry. We will in particular cover :
-- Main examples of commutative rings : polynomials, formal series, rings of integers
-- Divisibility, factorizations, ideals in commutative rings. Arithmetic applications.
-- Surgery on commutative rings : localization, tensor products. Geometric interpretation.
-- Modules of finite type over principal rings, and applications to matrices.
-- Extensions of fields. Galois theory.
Time permitting, we will also introduce the basic ideas of the language of categories, and maybe a bit of homological algebra.