A Dedekind domain is a UFD iff its ideal class group is trivial

Let $𝑅$ be a Dedekind domain. Then $𝑅$ is a UFD iff the ideal class group $C l 𝑅$ is trivial. ring

Check

If the ideal class group is trivial, then all ideals are principal, so $𝑅$ is a PID and thus UFI translates to the UFD property.