Unique factorization and Bezout iff principal ideal
Statement
The following are equivalent for an integral domain:
- It is a principal ideal domain.
- It is both a unique factorization domain and a Bezout domain.
Related facts
Facts used
- PID implies UFD
- PID implies Bezout
- Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain
- Strictly multiplicatively monotone norm on Bezout domain is a Dedekind-Hasse norm
- Dedekind-Hasse norm implies principal ideal ring
Proof
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).