Does the arrow "$\twoheadrightarrow$" mean surjective? In other words, for instance, can I gain any additional information from the following statement other than $f$ being surjective:
If $M$ is an $R$-module for some arbitrary ring $R$, $P$ is a free $R$-module of finite type, and $f: M \twoheadrightarrow P$ is a surjective $R$-module homomorphism, then there exists an isomorphism $M \cong P \oplus \mathsf{Ker}(f)$.