# Principal ideal ring iff every prime ideal is principal

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

View a complete list of pages giving proofs of equivalence of definitions
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 a commutative unital ring:

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.

## Proof

### 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.