TITLE: What's the intuition behind a split exact sequence?
QUESTION [10 upvotes]: I've learned the following definition for a split exact sequence:
A short exact sequence of $R$-module homomorphisms $0\to A \stackrel{f}{\to} B \stackrel{g}{\to} C \to 0$ is split if there exists a homomorphism $\alpha:B\to A$ such that $\alpha\circ f = 1$ or a homomorphism $\beta:C\to B$ such that $g\circ \beta = 1$.
I can never remember whether the definition requires $\alpha\circ f = 1$ or $f \circ \alpha = 1$, and similarly for the other mapping, $\beta$. I suspect that my inability to remember this stems from an underlying lack of understanding about what split exact sequences are for, what a sequence being split really tells us about the modules and morphisms involved, and what concept split exact sequences are meant to capture or generalize.
Is there a good way to remember the directions of composition for these definitions?
What is the intuition, utility, and underlying meaning behind a split exact sequence?

REPLY [4 votes]: When you have a short exact sequence
$$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 $$
$B$ canonically has a submodule $\mathrm{im}(f) \cong A$ and a quotient module $\mathrm{coim}(f) \cong C$. When the sequence is split, we can actually write $B$ as a direct sum of these two modules, which lets us decompose the given short exact sequence into a direct sum of two short exact sequences:
$$
 \begin{matrix} 
0 \to A \xrightarrow{f} &B& \xrightarrow{g}  C \to 0 
\\ & \cong
\\ 0 \to A \xrightarrow{f} &\mathrm{im}(f)& \to0 \to 0 
\\ & \oplus
\\ 0 \to 0 \to & \mathrm{coim}(g) & \xrightarrow{g} C \to 0 
\end{matrix}$$
I imagine this is where the term split comes from.
A splitting $\beta$ of the epimorphism $g$ is meant to be the monic map expressing how $\mathrm{coim}(g)$ is being made into a submodule of $B$ in a way compatible with $g$. So $g \circ \beta = 1_C$.
Similarly for the splitting $\alpha$ of the monomorphism $f$.

Having a pair of morphisms
$$ Y \xrightarrow{u} X \xrightarrow{v} Y $$
with $v \circ u = 1_Y$ is a significant phenomenon going by lots of names coming from lots of mathematical settings. For example:

$v$ is a left inverse of $u$
$u$ is a right inverse of $v$
$u$ is a split monomorphism, with $v$ its splitting
$v$ is a split epimorphism, with $u$ its splitting
$u$ is a section of $v$
We say that $Y$ is a retract of $X$, given by the retraction $(u,v)$
$u \circ v$ is a split idempotent, with the splitting given by $(u,v)$