Principal Ideal Domain is Dedekind Domain
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Let $D$ be a principal ideal domain which is specifically not a field.
Then $D$ is a Dedekind domain.
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.