# Principal ideal ring iff every prime ideal is principal

From Commalg

This article gives a proof/explanation of the equivalence of multiple definitions for the term principal ideal ringView 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 domainView a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

The following are equivalent for a commutative unital ring:

- Every ideal in the ring is a principal ideal.
- Every 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.

## 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.