Lemma 38.7.6. Assumption and notation as in Lemma 38.7.1. Assume moreover that $N$ is projective as an $R$-module. Then each $s \in \Sigma$ defines a universally injective $R$-module map $s : N \to N$, and the map $N \to \Sigma ^{-1}N$ is $R$-universally injective.

Proof. Pick $s \in \Sigma$. By Lemma 38.7.5 the map $s : N \to N$ is universally injective. The map $N \to \Sigma ^{-1}N$ is universally injective as the directed colimit of the maps $s : N \to N$. $\square$

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).