# Math Help - Compute Hom(q,q/z)

1. ## Compute Hom(q,q/z)

Hom(z,q/z)=?
Hom(q,q/z)=?

and also please prove that Q is an injective Z-module.

2. ## Re: Compute Hom(q,q/z)

$Hom (\mathbb{Z} , \mathbb{Q} / \mathbb{Z} )$ is trivially isomorphic to $\mathbb{Q} / \mathbb{Z}$ since any morphism is determined by where it sends $1$. To prove that $\mathbb{Q}$ is injective take any morphism $f: n\mathbb{Z} \rightarrow \mathbb{Q}$ with $f(n)=a$ then, since $\mathbb{Q}$ is divisible there exists $b\in \mathbb{Q}$ with $nb=a$ take $g: \mathbb{Z} \rightarrow \mathbb{Q}$ with $g(1)=b$ then $gi=f$ where $i :n\mathbb{Z} \rightarrow \mathbb{Z}$ is the inclusion, conclude by Baer's criterion. I don't have the answer to the second but maybe the fact that $\mathbb{Q} / \mathbb{Z} \cong \oplus_{p \ prime } \mathbb{Z}_{p^\infty}$ is of use.

3. ## Re: Compute Hom(q,q/z)

thank you jose i got it. I will try for the others.