# UFD iff PID in Dedekind domain.

Let $A$ be a Dedekind domain. PID implies UFD. So for the other direction assume $A$ is an UFD. In this proof the author only considers prime ideals instead of any proper ideal. Why is this sufficient?

• In the case where $A$ is Dedekind every ideal is a product of prime ideals and the product of principal ideals is principal. May 15 '18 at 21:05