I have two exercises which are driving me crazy.

1) Is Q finitely generated Z-module?

2) Let $\displaystyle d,e \in N/\{0\}$. Show, that $\displaystyle Hom_Z(Z_d,Z_e) \cong Z_f$, where $\displaystyle f = gcd (d,e)$.

I have previously proved that $\displaystyle Hom_Z(Z_d, G) \cong \{x \in G | dx = 0\}$, where G is an Abelian group and $\displaystyle d \in N/\{0\}$, so it's enough to show that $\displaystyle H:=\{\overline{n} \in Z_e | d\overline{n} = 0\} \cong Z_f$. And I have been given the rule $\displaystyle \overline{k} \mapsto \overline{k(e/f)}$. So I have to show that $\displaystyle f: Z_f \rightarrow H$, $\displaystyle f(\overline{k}) = \overline{k(e/f)}$, is isomorphism.

I have been able to show that this function is homomorphism and injective, but the problem here is how I can show that this function is surjective?