I was reading a section on Projective modules, that asked us to prove that a projective module is a direct summand of a free module. I then began thinking whether every module should be a direct summand of a free module.
From what I understand, a module $M$ is a summand of a free module $F$ if $M\oplus M'=F$ for some module $M'$. I think of a module $M$ as the free module $F$ generated by its generators, which is then quotiented by the relations between those generators. Those relations form a submodule $N$. Hence, $M=F/N$. In other words, $F=M\oplus N$. Am I understanding this correctly?