Necessary and sufficient condition for projective modules

Let us have a $P$ module. We have to prove the following statement:

$P$ is projective $\Leftrightarrow$ There is an $F$ free module, such that $F \cong F \oplus P$.

I have already seen the $\Rightarrow$ direction in a previously asked question, but can't really think of anything to prove the other way.

Any help appreciated!

• If $F\cong F\oplus P$, then $P$ is a direct summand of a free module, which is one of the usual characterizations of a projective module. – Ben West Sep 13 '17 at 14:49

If $F\cong F\oplus P$, then $P$ is a direct summand of a free module, which is one of the usual characterizations of a projective module. – Ben West 16 mins ago
• A stupid (!) question: if $F \cong F\oplus P$, how the ranks add up? Shouldn't it be "there exists a module $Q$ such that $Q\oplus P$ is a free module of finite rank"? – Krish Sep 13 '17 at 15:49