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Let $X$ be a locally Noetherian scheme with finite number of associated points. Let $f$ be a function on $X$, that is $f$ is a global section of the structure sheaf, if $f$ vanishes at one associated point, is $f$ a zerodivisor?

If $X$ is an affine Noetherian scheme, $f$ is a zerodivisor. Is it right when $X$ is not affine?

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It's a good question and I thought of it some months ago, but I didn't reach the solution. However, I'd like to recall that the sheaf $f \mathcal{O}_X$ is invertible iff $f$ does not vanish at any associated point. Maybe it could be useful. –  Andrea Aug 23 '11 at 15:22
It seems this is not the case if $X$ has only one associated point. –  Rodrigo Jan 31 at 13:18
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