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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 $M \cong P \oplus \mathsf{Ker}(f)$.

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It can mean all sort of things, of course. Usually, it means some variation of surjective (like admissible surjection, or conflation or...) In the statement you quote, though, it simply means surjection.

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