Lemma 10.30.4. Let $R$ be a domain. Let $\varphi : R \to S$ be a ring map. The following are equivalent:

The ring map $R \to S$ is injective.

The image $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ contains a dense set of points.

There exists a prime ideal $\mathfrak q \subset S$ whose inverse image in $R$ is $(0)$.

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