Principal ideal ring iff every prime ideal is principal

Statement
The following are equivalent for a commutative unital ring:


 * Every ideal in the ring is a fact about::principal ideal.
 * Every fact about::prime ideal in the ring is a principal ideal.

In particular, a principal ideal ring can be defined as a commutative unital ring in which every prime ideal is principal, and a principal ideal domain can be defined as an integral domain in which every prime ideal is maximal.

Every ideal is principal implies every prime ideal is principal
The proof of this is tautological.

Every prime ideal is principal implies every ideal is principal
Here is the proof outline:


 * If the collection of non-principal ideals under inclusion is nonempty, it satisfies the conditions for Zorn's lemma. In particular, if there exist non-principal ideals, there exists an ideal maximal with respect to being non-principal.
 * If there exists an ideal maximal with respect to being non-principal, it must be a prime ideal.