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

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

    Please compute;
    Hom(z,q/z)=?
    Hom(q,q/z)=?

    and also please prove that Q is an injective Z-module.
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  2. #2
    Super Member
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    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.
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  3. #3
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    Re: Compute Hom(q,q/z)

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