What is a typical example of the tensor product of modules failing to be left exact?

Question

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.

Answer 1

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.