Unique factorization and Dedekind iff principal ideal

This article gives a proof/explanation of the equivalence of multiple definitions for the term principal ideal domain

View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for an integral domain:

• It is a unique factorization domain as well as a Dedekind domain.
• It is a principal ideal domain.

Facts used

1. PID implies UFD
2. PID implies Dedekind
3. Dedekind implies one-dimensional
4. Unique factorization and one-dimensional iff principal ideal

Proof

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).