Noetherian and Bezout iff principal ideal

Statement

A ring is a principal ideal ring if and only if it is both a Noetherian ring and a Bezout ring.

In particular, an integral domain is a principal ideal domain if and only if it is both a Noetherian domain and a Bezout domain.

Definitions used

A ring is a:

• principal ideal ring if every ideal is a principal ideal
• Noetherian ring if every ideal is a finitely generated ideal
• Bezout ring if every finitely generated ideal is a principal ideal

Proof

The fact that a Noetherian Bezout ring is a principal ideal ring follows by a single line of logic based on the above definitions. The fact that every principal ideal ring is Bezout is again direct from the definitions. The fact that it is Noetherian follows from the fact that a principal ideal is, by definition, finitely generated.