The Stacks project

Lemma 10.160.2. Let $R$ be a Noetherian complete local ring. Any quotient of $R$ is also a Noetherian complete local ring. Given a finite ring map $R \to S$, then $S$ is a product of Noetherian complete local rings.

Proof. The ring $S$ is Noetherian by Lemma 10.31.1. As an $R$-module $S$ is complete by Lemma 10.97.1. Hence $S$ is the product of the completions at its maximal ideals by Lemma 10.97.8. $\square$

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