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Is this true?

Every UFD is a PID

No — it is false.

The counterexample

This one comes with the Notes pack.

See pricing

85 counterexamples are free — including most of the ones Part C leans on hardest.

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
rings

Others that fail the same way

From Rings and Fields › Euclidean, PID, UFD hierarchy

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