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
- PID implies UFD
- PID implies Dedekind
- Dedekind implies one-dimensional
- 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).