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

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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• $\mathbb{Q}$ is an injective $\mathbb{Z}$-module.
• $\mathbb{Z}$ is a Noetherian ring.