TITLE: Prove that every module $M$ is epic image of a projective module.
QUESTION [5 upvotes]: My Proof: We knew that every module $M$ is epic image of a free module and a free module is a projective module, hence every module $M$ is epic image of a projective module.
Is that true?! If false, please give me a hint.

REPLY [2 votes]: This CW answer intends to remove the question from the unanswered queue.

As already noted in the comments, your proof is right.