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    $\begingroup$ Martin- This is a history question, and yet your answer doesn't cite any sources. You seem to just be saying it's equally possible that the term came from differential geometry. $\endgroup$
    – Ben Webster ♦
    Commented Apr 20, 2010 at 16:58
  • $\begingroup$ Right . $\endgroup$
    – Martin Brandenburg
    Commented Apr 20, 2010 at 17:10
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    $\begingroup$ @Martin: localizing at a multiplicative set containing zero divisors (as in your example!) was not introduced until Uzkov around 1940's; even Chevalley in 1943 didn't use such generality, according to Emerton's reference. Your belief that such an equality was known "long before Zariski" seems to be a wild guess, moreover contrary to the historical evidence on the algebra side. Is there any evidence for it? Why would pre-Zariski geometers or function theorists even want such a result? The ring of germs is a quotient of $C^{\infty}(M)$, so why would they bother with abstract localization? $\endgroup$
    – Boyarsky
    Commented Jun 29, 2010 at 11:11