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Any projective module is a direct sum of countably generated projective modules.

Theorem 10.84.5. If $P$ is a projective $R$-module, then $P$ is a direct sum of countably generated projective $R$-modules.

Proof. A module is projective if and only if it is a direct summand of a free module, so this follows from Theorem 10.84.4. $\square$

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Comment #1337 by Hua Wang on

Suggested slogan: Any project module is a direct sum of countably generated projective modules

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