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Let $M$ be a module over a commutative ring $R$. Let $s \in M$ be an element such that for any $x \in \mathrm{Spec}\,R$, the image of $s$ in $M \otimes \kappa(x)$ is 0 (where $\kappa(x)$ is the residue field at $x$). What is the most natural necessary (and may be even sufficient) condition on M that allows to conclude that $s=0$? E.g. if $R$ is reduced and $M$ is (locally) free then this holds, but I can think of many other examples and non-examples.

update: obvious non-example is a non-reduced ring viewed as a module over itself. If one takes a ring, even reduced, then $M=R/I$ is a non-example, if $R/I$ is a non-reduced ring, and an example if it is a reduced ring. Of course, taking direct sums produces more (non-)examples.

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Dear Dima, could you please tell us about a few of your many (non)-examples? –  Georges Elencwajg Apr 26 '12 at 8:26
    
Dear Georges, I have updated the question. –  Dima Sustretov Apr 26 '12 at 9:09
    
Thank you, Dima. –  Georges Elencwajg Apr 26 '12 at 13:30

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