Let $A \subset B \subset C $ be rings. Suppose that $A$ is Noetherian and $C$ is finitely generated as an $A$-algebra. I want to show that $C$ is finitely generated as a $B$-module $ \iff $ $C$ is integral over $B$.
I have the following propositions:
Proposition 5.1: The following are equivalent for rings B $\subset$ C
i) $x \in C $ is integral over B
ii) B[x] is a finitely generated B-module
iii) B[x] is contained in a subring C' of C at C is a finitely generated B-module
Let $x_1, ... x_n \in C $ be integral over B. Then the ring $B[x_1, ... x_n] $ is a finitely-generated B-module.