# Z modules and exact sequences

• Jan 15th 2013, 10:27 AM
I-Think
Z 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 AM
Curu
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.
• Jan 16th 2013, 08:43 AM
johng
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.
Attachment 26582