A short exact sequence is pure

Question

I have the next doubt about the equivalences of a short sequence is pure.

The definition is: Let $0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0$ a short exact sequence of abelian groups whith morphisms $\mu:A'\rightarrow A$ and $\epsilon:A\rightarrow A''$.

The sequence is pure if, whenever $\mu(a')=ma$, $a'\in A'$, $a\in A$, $m$ a positive integer, then there exist $b'\in A'$ such that $a'=mb'$.

I need to prove the next two statements are equivalents:

$i)$ The sequence is pure;

$ii)$ Given $a''\in A''$ with $ma''=0$, there exist $a\in A$ with $\epsilon(a)=a''$, $ma=0$.

I proved $ii)\Longrightarrow i)$ for the other direction this is what I have:

Let $a''\in A''$ with $ma''=0$, since $\epsilon$ is surjective, it follows that $a''=\epsilon(a)$ for some $a\in A$.

Then we have that $\epsilon(ma)=0$, by exactness of the sequence follows that $\mu(a')=ma$ and by hypothesis $a'=mb'$ .

I do not know how to follow from here so, any hint would be appreciated.

Thank you!

Answer 1

You're almost there.

Consider $a-\mu(b')$.