If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is exact and $B\simeq A\oplus C$ as a $R$-module, does this sequence split?

Question

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?

Please explain what you thought about, and generally try to follow this guide: http://meta.math.stackexchange.com/a/1804/26369 — Mark S., Nov 23 '13 at 17:16
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 . — kpax, Nov 23 '13 at 17:27

score 0 · Accepted Answer · edited Apr 13 '17 at 12:20

0

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 $B \cong A \oplus C$.

edited Apr 13 '17 at 12:20

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answered Nov 23 '13 at 17:35

Mark S.

25,893

If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is exact and $B\simeq A\oplus C$ as a $R$-module, does this sequence split?

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