In this work we establishe relations between modules over a ring and vector spaces, we show results of linear algebra that can be extended to modules and we present counterexamples to those ones that cannot be extended. By identify abelian groups with Z-modules we classify every the finitely generated abelian groups and we show a decomposition in direct sum of ciclic subgroups. Finally, we apply the results about finitely generated abelian groups to determine the rational canonical form of an endomorphis of finitely generated K[t]-modules and simplify the computer of associated numbers to the endomorphism, for example: therank and the determinant.