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

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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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