# The Eilenberg swindle

If $M$ is a projective $A$-module, there exists a free $A$-module $F$ such that $F \simeq M \bigoplus F$.

Let $N$ and $L$ be $A$-modules such that $L \cong M \bigoplus N$ is free. Then $F = \bigoplus_{n \in \mathbb{N}}L$ is free and $M \bigoplus F \cong M \bigoplus_{n \in \mathbb{N}}(M \bigoplus N) \cong (\bigoplus_{n \in \mathbb{N}_0}M) \bigoplus (\bigoplus_{n \in \mathbb{N}}N) \cong (\bigoplus_{n \in \mathbb{N}}M) \bigoplus (\bigoplus_{n \in \mathbb{N}}N) \cong \bigoplus_{n \in \mathbb{N}}(M \bigoplus N) \cong F$.
2. More generally, if $P,P'$ are projective modules, there exists some module $Q$ such that $P\oplus Q\simeq P'\oplus Q$ is free (the original problem is the particular case where $P'=0$). The proof of the generalization is essentially the same.