Lemma 10.92.4. 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.
By Lemmas 10.81.7 and 10.88.7, $M$ is flat and Mittag-Leffler, so the conclusion follows from Theorem 10.92.3.
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