ABSTRACT
At the end of section 10 we started investigating extensions of commutative rings R ⊆ S where S is obtained from R by adjoining a finite number of elements s 1, …, s n so that S is a finitely generated R-algebra. Since the surjective homomorphism Φ : R[X 1, …, X n ] →; R[s 1, …, s n ] given by p ↦ p(s 1, …, s n ) induces an isomorphism R[s 1, …, s n ] ≅ R[X 1, …, X n ]/I where I : = ker Φ = {p ∈ R[X 1, …, X n ] | p(s 1, …, s n ) = 0}, the class of finitely generated R-algebras consists exactly of all homomorphic images of the polynomial rings R[X 1, …, X n ]. Therefore, it is very natural in the context of polynomial rings to also study finitely generated algebras.
