I have a question about my method of proof for proving a simple fact about projective modules. I have a feeling my idea is wrong and I was hoping some one could point out where the mistake is.
Let $R$ be a commutative ring with identity.
Let $P$ be a projective $R$-module. Prove that there exists a free $R$-module $F$ such that $P \oplus F \cong F$.
Sketch of Proof: Since $P$ is projective there exits a free $R$-module $F$ such that $F=P \oplus M$ for some module $M$. Then we can consider the direct sum $P \oplus F \cong P \oplus P \oplus M$. Thus it suffices to check $P \oplus P \cong P$. Is the last isomorphism justifiable because P is projective? Am I on the right track or should I try something else?