Please compute;
Hom(z,q/z)=?
Hom(q,q/z)=?
and also please prove that Q is an injective Z-module.
is trivially isomorphic to since any morphism is determined by where it sends . To prove that is injective take any morphism with then, since is divisible there exists with take with then where is the inclusion, conclude by Baer's criterion. I don't have the answer to the second but maybe the fact that is of use.