Krull-Azikuzi theorem

Statement

Let:

• $R$ be a Noetherian integral domain of Krull dimension 1
• $K$ be the field of fractions of $R$
• $L$ be a finite extension field of $K$
• $S$ be a subring of $L$ that contains $R$

Then the following hold:

• $S$ is Noetherian
• The Krull dimension of $S$ is at most 1
• Given any nonzero ideal of $R$, there are only finitely many ideals of $S$ containing that

In particular, the integral closure of $R$ in $L$ is Noetherian