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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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    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)2011-05-16

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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.