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Is this true?

Every prime ideal is maximal

No — it is false.

The counterexample

(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.

The kind of mistake this is

Standard counterexample

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

Drill statements like this

Others that fail the same way

From Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT

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