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?