Lemma 10.164.2. Let $R \to S$ be a ring map. Assume that

$R \to S$ is faithfully flat, and

$S$ is reduced.

Then $R$ is reduced.

Lemma 10.164.2. Let $R \to S$ be a ring map. Assume that

$R \to S$ is faithfully flat, and

$S$ is reduced.

Then $R$ is reduced.

**Proof.**
This is clear as $R \to S$ is injective.
$\square$

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## Comments (1)

Comment #873 by Dylan on