Lemma 10.16.9. Let $\varphi : R \to S$ be a ring map. Let $\mathfrak p$ be a prime of $R$. The following are equivalent

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

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

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

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

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

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