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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