Artin-Tate lemma

Statement

Suppose $A,B,C$ are commutative unital rings, such that:

• $A$ is Noetherian
• $B$ and $C$ are $A$-algebras, with $B$ a subalgebra of $C$
• $C$ is finitely generated as an $A$-algebra
• $C$ is finitely generated as a $B$-module

Then: $B$ is finitely generated as an $A$-algebra