I was reading "Serial Rings" by Gennadi Puninski. There it is written that , "Since every module is a homomorphic image of a free module, every projective module is a direct summand of free module".(ie. if $P$ is a projective module, there exists a free module F such that, $ F=P \oplus T$ for some module $T$.)
But I can't understand how "Every module is a homomorphic image of a free module" implies that "Every projective module is a direct summand of free module".
(I have found a proof for "Every projective module is a direct summand of free module" but the first part of the above mentioned sentence wasn't used there.)