# Example of a particular short exact sequence that does not split.

Can somebody give me an example of a short exact sequence of $Z$-modules that starts from $\oplus{\mathbb{Q}_{i=1}^\infty}$ and does not split?

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## 1 Answer

This seems not to be possible, see this nlab article on injective modules. We have that:

• $\mathbb{Q}$ is an injective $\mathbb{Z}$-module.
• $\mathbb{Z}$ is a Noetherian ring.
• For a Noetherian ring any direct sum of injective modules is again injective.
• If a module is injective, then every short exact sequence starting in it will split.
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