Suppose we have a short exact sequence.
$0 \to A \to B \to C \to 0$.
Suppose also that we know there is some embedding $e : C \hookrightarrow B$.
Can we conclude that the sequence splits? It seems that we should be able to, but it is not required that $e$ be a section of the surjection $B \to C$.
Any help is appreciated.
There is no reason for this to hold. For example, there is a short exact sequence $$0\rightarrow \mathbb{Z}/2\mathbb{Z}\rightarrow \mathbb{Z}/4\mathbb{Z}\rightarrow\mathbb{Z}/2\mathbb{Z}\rightarrow 0$$ but it doesn't split since $\mathbb{Z}/4\mathbb{Z}$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$. And even a sequence with $B\cong A\oplus C$ need not split in general, as can be seen in this question.
