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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