Definiton : Let $M$ be an $A$-module. Then $M$ is projective if there exists an $A$-module $N$ such that $M \oplus N$ is free.
Prop: If $M$ is free then $M$ is projective.
Can we simply take the trivial module $\{0\}$ then $M \oplus \{0\} \cong M$ is free, so $M$ is projective?