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.

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

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05A0. Beware of the difference between the letter 'O' and the digit '0'.