# Thread: Z modules and exact sequences

1. ## Z modules and exact sequences

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

2. ## Re: Z modules and exact sequences

$0 \to \mathbb Z \to\mathbb Q \to \mathbb{Q/Z} \to 0$. 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.

3. ## Re: Z modules and exact sequences

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