Counterexample

Is this true?

Every integral domain is a UFD

No — it is false.

The counterexample

$\mathbb{Z}[\surd-5]: 6 = 2\cdot3 = (1+\surd-5)(1-\surd-5)$

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Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Go deeper· Explained in full in The ring hierarchy and where each inclusion is strict· Asked in CSIR NET December 2024· From Rings and Fields › Euclidean, PID, UFD hierarchy

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

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