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

Let $R$ be a [[Dedekind domain]].
Then $R$ is a [[Unique factorization domain|UFD]] iff the [[ideal class group]] $\Cl R$ is trivial. #m/thm/ring 

> [!check]-
> If the ideal class group is trivial, then all ideals are principal, so $R$ is a [[Principal ideal domain|PID]] and thus [[Unique factorization of ideals|UFI]] translates to the [[Unique factorization domain|UFD]] property. <span class="QED"/>


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