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 $s : N \to N$.
$\square$

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