Lemma 10.17.9. Let $\varphi : R \to S$ be a ring map. Let $\mathfrak p$ be a prime of $R$. The following are equivalent

1. $\mathfrak p$ is in the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$,

2. $S \otimes _ R \kappa (\mathfrak p) \not= 0$,

3. $S_{\mathfrak p}/\mathfrak p S_{\mathfrak p} \not= 0$,

4. $(S/\mathfrak pS)_{\mathfrak p} \not= 0$, and

5. $\mathfrak p = \varphi ^{-1}(\mathfrak pS)$.

Proof. We have already seen the equivalence of the first two in Remark 10.17.8. The others are just reformulations of this. $\square$

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