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Suppose $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact sequence that $B\simeq A\oplus C$ as a $R$-module. Does this short exact sequence split?

I think the answer is no, but I must have a counterexample. I could not find any. Can you help me?

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  • $\begingroup$ Please explain what you thought about, and generally try to follow this guide: meta.math.stackexchange.com/a/1804/26369 $\endgroup$
    – Mark S.
    Nov 23, 2013 at 17:16
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    $\begingroup$ I tried to make a short exact sequence that doesnt split but $B\simeq A\oplus C$ !I used $\mathbb{Q}$ and $\mathbb{Z}$ because the only homomorphism between them is zero but I couldnt make the hypothesis . $\endgroup$
    – kpax
    Nov 23, 2013 at 17:27

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Although this isn't quite a duplicate, the case $R=\mathbb{Z}$ is thoroughly discussed in A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$.

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