= Vanishing criterion in a module localization
{title2=$m/1=0\text{ in }S^{-1}M\iff sm=0\text{ for some }s\in S$}
The fraction relation defining <localization of a module> gives this criterion. For localization at powers of a single element $f$, a fraction vanishes exactly when some power of $f$ annihilates its numerator. Finitely many such vanishing relations can be cleared with one common exponent. This is distinct from detecting global zero by localizing at every prime ideal.
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