The Stacks project

Proof. It is immediate from the definitions that (1) is equivalent to (4) by looking at part (5) of Lemma 10.26.5. Recall that $\mathop{\mathrm{Spec}}(S)$ is sober, Noetherian, and Jacobson, see Lemmas 10.26.2, 10.31.5, 10.35.2, and 10.35.4. If $S$ has dimension $0$, then every point defines an irreducible component and there are only a finite number of irreducible components (Topology, Lemma 5.9.2). Thus (1) implies (2). Trivially (2) implies (3). If (3) holds, then $\mathop{\mathrm{Spec}}(S)$ is discrete by Topology, Lemma 5.18.6 and hence the dimension of $S$ is $0$. $\square$