Let $0 \rightarrow M' \xrightarrow{f} M \xrightarrow{g} M'' \rightarrow 0$ be an exact sequence of modules on a commutative unitary ring.
We say that the exact sequence is split when M can be written as the direct sum of $M_1$ and $M_2$ such that $f$'s corestriction to $M_1\oplus \lbrace 0 \rbrace$ is an isomorphism and $g$'s restriction to $\lbrace 0 \rbrace \oplus M_2$ is also one.
Is it correct to say that it is split if and only if $M \simeq \text{Im}f \oplus M/\text{Im}f$ ? I think this property is always true of vector spaces and probably free modules, and I know that this kind of exact sequence is always split when it involves vector spaces.
I ask this because $f$'s corestriction to $\text{Im}f$ is obviously an isomorphism since $f$ is injective, and because $\text{Im}f=\text{ker}g$, $g$ is actually a function on the supplementary submodule alone.