Lemma 10.108.3 (04PT)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.108: Pure ideals
Lemma 10.108.3 (cite)

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Pure ideals are determined by their vanishing locus.

Lemma 10.108.3. Let $R$ be a ring. If $I, J \subset R$ are pure ideals, then $V(I) = V(J)$ implies $I = J$ .

Proof. For example, by property (7) of Lemma 10.108.2 we see that $I = \mathop{\mathrm{Ker}}(R \to \prod _{\mathfrak p \in V(I)} R_{\mathfrak p})$ can be recovered from the closed subset associated to it. $\square$

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Comment #1221 by David Corwin on December 28, 2014 at 03:30

Suggested slogan: Pure ideals are determined by their vanishing locus

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