Lemma 15.47.4. Let $R \to S$ be a ring map. Assume that

1. $R$ is a Noetherian domain,

2. $R \to S$ is injective and of finite type, and

3. $S$ is a domain and J-0.

Then $R$ is J-0.

Proof. After replacing $S$ by $S_ g$ for some nonzero $g \in S$ we may assume that $S$ is a regular ring. By generic flatness we may assume that also $R \to S$ is faithfully flat, see Algebra, Lemma 10.118.1. Then $R$ is regular by Algebra, Lemma 10.164.4. $\square$

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