Is the following statement true? Every prime ideal is maximal

No. (X) in ℤ[X], or (0) in any integral domain that is not a field ℤ[X]/(X) ≅ ℤ is a domain but not a field. In a PID the implication does hold for non-zero primes.

Counterexample

Is this true?

Every prime ideal is maximal

No — it is false.

The counterexample

(X) in $\mathbb{Z}[X]$ , or (0) in any integral domain that is not a field

Why it fails

$\mathbb{Z}[X]/(X) \cong \mathbb{Z}$ is a domain but not a field. In a PID the implication does hold for non-zero primes.

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Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Every integral domain is a UFD
Every UFD is a PID
Every PID is Euclidean

See real exam options built on this mistake Drill statements like this

rings

Go deeper· Explained in full in Ideals, quotients and the Chinese remainder theorem· Asked in CSIR NET December 2025· From Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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