Lemma 15.128.2. Let $R$ be a ring. Let $P$ be a projective module. There exists a free module $F$ such that $P \oplus F$ is free.

Proof. Since $P$ is projective we see that $F_0 = P \oplus Q$ is a free module for some module $Q$. Set $F = \bigoplus _{n \geq 1} F_0$. Then $P \oplus F \cong F$ by Lemma 15.128.1. $\square$

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