The Stacks project

Lemma 10.93.4. Let $f: M \to N$ be universally injective map of $R$-modules. Suppose $M$ is a direct sum of countably generated $R$-modules, and suppose $N$ is flat and Mittag-Leffler. Then $M$ is projective.

Proof. By Lemmas 10.82.7 and 10.89.7, $M$ is flat and Mittag-Leffler, so the conclusion follows from Theorem 10.93.3. $\square$

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