Unique factorization and Bezout iff principal ideal

Statement
The following are equivalent for an integral domain:


 * It is a fact about::principal ideal domain.
 * It is both a fact about::unique factorization domain and a fact about::Bezout domain.

Related facts

 * Unique factorization and Dedekind iff principal ideal
 * Bezout and Dedekind iff principal ideal

Facts used

 * 1) uses::PID implies UFD
 * 2) uses::PID implies Bezout
 * 3) uses::Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain
 * 4) uses::Strictly multiplicatively monotone norm on Bezout domain is a Dedekind-Hasse norm
 * 5) uses::Dedekind-Hasse norm implies principal ideal ring

Principal ideal domain implies unique factorization and Bezout
This follows from facts (1) and (2).

Unique factorization and Bezout implies principal ideal domain
This follows by combining facts (3), (4), and (5).