Vanishing criterion in a module localization ($m/1=0\text{ in }{S}^{-1}M⟺sm=0\text{ for some }s\in S$)

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.