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

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

Facts used

1. PID implies UFD
2. PID implies Bezout
3. Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain
4. Strictly multiplicatively monotone norm on Bezout domain is a Dedekind-Hasse norm
5. 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).

References

Textbook references

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