Find an example of a short exact sequence of Z-modules that does not split.

Any hints?

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- Jan 15th 2013, 10:27 AMI-ThinkZ modules and exact sequences
Find an example of a short exact sequence of Z-modules that does not split.

Any hints? - Jan 16th 2013, 07:40 AMCuruRe: Z modules and exact sequences
$\displaystyle 0 \to \mathbb Z \to\mathbb Q \to \mathbb{Q/Z} \to 0$. It is clearly a short exact sequence of abelian group and $\displaystyle \text{Hom}_\mathbb{Z}(\mathbb{Q/Z}, \mathbb{Q})=0$, since $\displaystyle \mathbb{Q/Z}$ is torsion while $\displaystyle \mathbb Q$ is torsion-free.

- Jan 16th 2013, 08:43 AMjohngRe: Z modules and exact sequences
The above answer was posted as I was writing the following different example, but I'll post it anyway.

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