I am trying to prove that for any divisible torsion $\mathbb{Z}$-module $V$,
this map 
$$f:\mathbb{Q}/\mathbb{Z}\otimes\_E\text{Hom}(\mathbb{Q}/\mathbb{Z},V)\longrightarrow V\mbox{ defined by }
f((q+\mathbb{Z})\otimes g)=g(q+\mathbb{Z})$$
is an isomorphism, where tensor is taken over the ring $E=\operatorname{End}(\mathbb Q/\mathbb Z)$.


It is easy to prove that $f$ is a homomorphism, but I couldn't prove that $f$ is bijective. Are there any special properties for divisible torsion $\mathbb{Z}$-modules that help in proving that the above map $f$ is bijective?



Divisible torsion $\mathbb{Z}$-modules
First, consider an element $v\in V$. As $V$ is torsion we can choose $n$ such that $n!v=0$. For $k\leq n$ put $u\_k=(n!/k!)v$. Then choose $u\_k$ for $k>n$ inductively with $ku\_k=u\_{k-1}$ (which is possible because $V$ is divisible). There is then a unique homomorphism $\phi\colon\mathbb{Q}/\mathbb{Z}\to V$ such that $\phi([1/k!])=u\_k$ for all $k$, and $f([1/n!]\otimes \phi)=v$. This shows that $f$ is surjective.


Now consider an element $\alpha\in\ker(f)$. As $\mathbb{Q}/\mathbb{Z}$ is the union of the cyclic subgroups generated by elements of the form $[1/n!]$, we see that $\alpha$ can be written as $[1/n!]\otimes\phi$ for some $\phi\colon\mathbb{Q}/\mathbb{Z}\to V$ with $\phi([1/n!])=0$. Now multiplication by $n!$ gives a surjective endomorphism of $\mathbb{Q}/\mathbb{Z}$, whose kernel is generated by $[1/n!]$. It follows easily that $\phi=n!\psi$ for some $\psi$, and thus that $\alpha=n![1/n!]\otimes\psi=0$. Thus, $f$ is also injective. 


The above argument shows that the composite
$$ \mathbb{Q}/\mathbb{Z} \otimes\_{\mathbb{Z}} \text{Hom}(\mathbb{Q}/\mathbb{Z},V) \to 
 \mathbb{Q}/\mathbb{Z} \otimes\_{\text{End}(\mathbb{Q}/\mathbb{Z})} \text{Hom}(\mathbb{Q}/\mathbb{Z},V) \to V
$$
is an isomorphism, and it follows easily that both of the maps involved are isomorphisms. One can also see more directly that the first map is an isomorphism, using the standard fact that $\text{End}(\mathbb{Q}/\mathbb{Z})$ is the profinite completion of the integers, together with the obvious fact that the first tensor factor $\mathbb{Q}/\mathbb{Z}$ is a torsion group.


