Principal Ideal Domain is Dedekind Domain

Theorem
Let $A$ be a principal ideal domain, which is not a field.

Then $A$ is a Dedekind domain.

Proof
By definition of principal ideal domain $A$ is an integral domain.

By Principal Ideal Domain is Noetherian $A$ is noetherian.

By Principal Ideal Domain is Integrally Closed $A$ is integrally closed.

By Prime Ideal of Principal Ideal Domain is Maximal $A$ has Krull dimension $\le 1$.

By Field has Dimension Zero and since $A$ is not a field, the Krull dimension of $A$ is $1$.

By definition of Dedekind domain $A$ is a Dedekind domain.