Exercise 6, § 3

An integral domain $A$ is called integrally Noetherian if it is Noetherian and if, for every finite sequence $(t_i)_{1 \leq i \leq n}$ of elements of an extension field of the field of fractions of $A$, the integral closure of $A[t_1, \ldots, t_n]$ is a finitely generated $A[t_1, \ldots, t_n]$-module. Every finitely generated integral algebra over a field is an integrally Noetherian domain (no. 2, Theorem 2).

(a) Every finitely generated integral algebra over an integrally Noetherian domain is an integrally Noetherian domain. Every ring of fractions of an integrally Noetherian domain not reduced to 0 is an integrally Noetherian domain.

(b) Let $A$ be an integrally Noetherian domain $(t_i)_{1 \leq i \leq n}$ a finite sequence of elements of an extension field of the field of fractions $K$ of $A$ and $B$ a subring of $K(t_1, \ldots, t_n)$ containing $A$ and integral over $A$. Then $B$ is a finitely generated $A$-module. (Note that the field of fractions of $B$ is a finite algebraic extension of $K$, using Algebra, Chapter V, § 5, no. 7.)