The Stacks project

Proof. Let $R \to S$ be a finite ring map. By Lemma 10.153.3 it suffices to show that $S$ is a product of local rings. By Lemma 10.36.21 $S$ has finitely many primes $\mathfrak m_1, \ldots , \mathfrak m_ r$ which all lie over $\mathfrak m$. There are no inclusions among these primes, see Lemma 10.36.20, hence they are all maximal. Every element of $\mathfrak m_1 \cap \ldots \cap \mathfrak m_ r$ is nilpotent by Lemma 10.17.2. It follows $S$ is the product of the localizations of $S$ at the primes $\mathfrak m_ i$ by Lemma 10.53.5. $\square$