Lemma 15.128.3. Let $R$ be a ring. Let $P$ be a projective module. Let $s \in P$. There exists a finite free module $F$ and a finite free direct summand $K \subset F \oplus P$ with $(0, s) \in K$.

**Proof.**
By Lemma 15.128.2 we can find a (possibly infinite) free module $F$ such that $F \oplus P$ is free. Then of course $(0, s)$ is contained in a finite free direct summand $K \subset F \oplus P$. In turn $K$ is contained in $F' \oplus P$ where $F' \subset F$ is a finite free direct summand.
$\square$

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