Theorem 10.92.3. Let $M$ be an $R$-module. Then $M$ is projective if and only it satisfies:

1. $M$ is flat,

2. $M$ is Mittag-Leffler,

3. $M$ is a direct sum of countably generated $R$-modules.

Proof. First suppose $M$ is projective. Then $M$ is a direct summand of a free module, so $M$ is flat and Mittag-Leffler since these properties pass to direct summands. By Kaplansky's theorem (Theorem 10.83.5), $M$ satisfies (3).

Conversely, suppose $M$ satisfies (1)-(3). Since being flat and Mittag-Leffler passes to direct summands, $M$ is a direct sum of flat, Mittag-Leffler, countably generated $R$-modules. Lemma 10.92.1 implies $M$ is a direct sum of projective modules. Hence $M$ is projective. $\square$

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