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:

• 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.

References

Textbook references

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