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If $R$ is a local ring and $M$ and $N$ are finitely generated R-modules such that $M\otimes N = 0$ then it follows from Nakayama's lemma that either $M=0$ or $N=0$. This I know. But now

I am looking for an example of a local ring $R$ and two modules $M,N$ such that $M\otimes N = 0$ but neither $M=0$ nor $N=0$.

Thanks

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Let $R$ be a domain which is not a field. If $K$ is the field of fractions of $R$. Then $K,K/R$ are non-zero $R$-modules satisfying $K\otimes_R(K/R)=0$. In particular, you could take $R$ local and have an example of the type sought.

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    @Kennan Kidwell: of course! Thanks.2012-12-08