Unique factorization and Dedekind iff principal ideal

Statement
The following are equivalent for an integral domain:


 * It is a fact about::unique factorization domain as well as a fact about::Dedekind domain.
 * It is a fact about::principal ideal domain.

Facts used

 * 1) uses::PID implies UFD
 * 2) uses::PID implies Dedekind
 * 3) uses::Dedekind implies one-dimensional
 * 4) uses::Unique factorization and one-dimensional iff principal ideal

Principal ideal implies UFD and Dedekind
This is facts (1) and (2).

UFD and Dedekind implies principal ideal domain
This is facts (3) and (4).