Principal Ideal Domain is Dedekind Domain

Theorem

Let $D$ be a principal ideal domain which is specifically not a field.

Then $D$ is a Dedekind domain.

Proof

By definition of principal ideal domain $D$ is an integral domain.

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

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

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

By Integral Domain has Dimension Zero iff Field and since $D$ is not a field, the Krull dimension of $D$ is $1$.

Hence $D$ is a Dedekind domain by definition.

$\blacksquare$