Kaplansky's theorem on projective modules

Kaplansky's theorem

Proof

First assume M is countably generated: we can write $M=\sum _{n=0}^{\infty }Ax_{n}$. Since M is projective, we can choose a module N such that $M\oplus N$ is free with a basis $u_{i},i\in I$.

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Now assume M is an arbitrary projective module. We