I have been working in a book on Homology by Hilton & Stammbach, wherein they introduce the idea of a "pure sequence of Abelian groups", which is a short exact sequence of Abelian groups
$$0\xrightarrow{} A'\xrightarrow{\mu} A\xrightarrow{\varepsilon} A''\xrightarrow{} 0$$
where if $\mu(a')=ma$ occurs then there must be an element $b'$ in $A'$ so that $a'=mb'$.
The authors ask as an exercise to show that pure sequences are precisely those that may be "reduced mod $m$" to arrive at another short exact sequence. I was able to show this in the general setting of what I am supposing would be called a "pure submodule", whose definition is obtained by taking the definition of a pure group sequence and allowing $m\in M$, where $M$ is the base ring for the $M-$modules $A', A, A''$(so that a pure group sequence is a pure module sequence with $M=\mathbb{Z}$).
Now in the section on the Hom($-,-$) functor, they are asking to show that pure group sequences are those so that Hom$(\mathbb{Z}_m,-)$ is exact when applied to the pure sequence.
I would like, if possible, to solve this problem in the more general setting of modules. To do that, I need an idea of how "$M_m$" should be defined. I am thinking that it should be $M/mM$, but I am not so sure.
Note: I am only looking for the generalization itself, not a proof of it(I aim to do that myself if such a generalization exists).