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I am looking for an example of an exact sequence of $R$-modules $$ 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 $$ and a $R$-module $N$, such that $$ 0 \rightarrow M' \otimes N \rightarrow M \otimes N \rightarrow M'' \otimes N \rightarrow 0 $$ fails to be exact.

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Have you tried coming up with one yourself? –  Qiaochu Yuan May 16 '11 at 9:21
    
@Qiaochu yes, I did. I didn't have the right idea, but maybe I should have tried harder. Many thanks to Henri in any case. –  Felix Hoffmann May 16 '11 at 10:43
    
In fact, I'd say it's a bit harder to produce examples where the sequence remains exact (provided you start with non-split sequences) –  Mariano Suárez-Alvarez May 16 '11 at 11:07

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The typical example is made up with $M=M'=\mathbb Z$, and $N= \mathbb Z/ n \mathbb Z$, with the injection $\mathbb Z \to \mathbb Z, x \mapsto n x$. When tensorized by $Id_{\mathbb Z/ n \mathbb Z}$, this map becomes zero.

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