Lemma 10.120.15. A PID is a Dedekind domain.
Proof. Let $R$ be a PID. Since every nonzero ideal of $R$ is principal, and $R$ is a UFD (Lemma 10.120.13), this follows from the fact that every irreducible element in $R$ is prime (Lemma 10.120.5) so that factorizations of elements turn into factorizations into primes. $\square$
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