Lemma 10.30.6. Let $R \to S$ be a ring map. The following are equivalent:

The kernel of $R \to S$ consists of nilpotent elements.

The minimal primes of $R$ are in the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$.

The image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is dense in $\mathop{\mathrm{Spec}}(R)$.

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