Lemma 10.9.4 (00CQ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.9: Localization
Lemma 10.9.4 (cite)

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Lemma 10.9.4. The localization $S^{-1}A$ is the zero ring if and only if $0\in S$ .

Proof. If $0\in S$ , any pair $(a, s)\sim (0, 1)$ by definition. If $0\not\in S$ , then clearly $1/1 \neq 0/1$ in $S^{-1}A$ . $\square$

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