The Stacks project

Lemma 15.10.2. Let $(A, I)$ be a Zariski pair. Then the map from idempotents of $A$ to idempotents of $A/I$ is injective.

Proof. An idempotent of a local ring is either $0$ or $1$. Thus an idempotent is determined by the set of maximal ideals where it vanishes, by Algebra, Lemma 10.23.1. $\square$

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