Lemma 26.12.3. An affine scheme $\mathop{\mathrm{Spec}}(R)$ is reduced if and only if $R$ is reduced.

**Proof.**
The direct implication follows immediately from Lemma 26.12.2 above. In the other direction it follows since any localization of a reduced ring is reduced, and in particular the local rings of a reduced ring are reduced.
$\square$

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