Let $M'$ be a submodule of $\mathbb{Z}$-module $M$, and let $i:M'\rightarrow M$ be a natural monomorphism.
How to prove the following theorem ? :
$i\otimes 1_N:M'\otimes_{\mathbb{Z}} N \rightarrow M\otimes_{\mathbb{Z}} N$ is a monomorphism for all $\mathbb{Z}$-modules $N$ iff for all $q\in \mathbb{Z}$ we have $M'\cap qM=qM'$.
If $N$ is finitely generated then we can use fact that $N$ is direct sum of cyclic modules and then we know that $\ker\left(M'\otimes \left(\mathbb{Z}/q\mathbb{Z}\right)\rightarrow M\otimes \left(\mathbb{Z}/q\mathbb{Z}\right)\right)\cong \left(M'\cap qM\right)/qM'$, so in this case the proof is easy. How to do it in general case ? Is it still true ?