Z modules and exact sequences

**I-Think** · January 15th 2013, 10:27 AM

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

**Curu** · January 16th 2013, 07:40 AM

$0 \to \mathbb Z \to\mathbb Q \to \mathbb{Q/Z} \to 0$ . It is clearly a short exact sequence of abelian group and $\text{Hom}_\mathbb{Z}(\mathbb{Q/Z}, \mathbb{Q})=0$ , since $\mathbb{Q/Z}$ is torsion while $\mathbb Q$ is torsion-free.

**johng** · January 16th 2013, 08:43 AM

The above answer was posted as I was writing the following different example, but I'll post it anyway.

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