The Stacks project

Proof. Let $\mathfrak p$ be a prime of $R$. By Lemma 10.17.9 we have see that the hypotheses of Topology, Lemma 5.12.10 are satisfied for the topological space $\mathop{\mathrm{Spec}}(R)$. Hence the connected component of $\mathfrak p$ in $\mathop{\mathrm{Spec}}(R)$ is the intersection of open and closed subsets of $\mathop{\mathrm{Spec}}(R)$ containing $\mathfrak p$. Hence it equals $V(I)$ where $I$ is generated by the idempotents $e \in R$ such that $e$ maps to $0$ in $\kappa (\mathfrak p)$, see Lemma 10.21.3. Any idempotent $e$ which is not in this collection clearly maps to $1$ in $R/I$. $\square$