Suppose we have two short exact sequences, $0\rightarrow B\rightarrow A\rightarrow C \rightarrow 0$ and $0\rightarrow C\rightarrow A\rightarrow B \rightarrow 0$.
Is there anything we can conclude about them? Do they necessarily split?
Alternatively, if $A/B\cong C$ and $A/C\cong B$, can we conclude that $A\cong B\times C$? If not, what are other needed conditions to make that conclusion?
Examples/counterexamples are appreciated.
Edit: A comment has pointed out that in general, no. But I'd love to get a comprehensive answer.