This question comes from a question in Passman's A course in Ring Theory. He asks to prove that a submodule of a free module, say $V$, over a von Neumann regular ring $R$ is flat. He says the reader should use the fact that $R$ is semihereditary, which I was able to prove. Now I know that finitely generated submodules of $V$ must be projective, hence flat, but am struggling to prove this for arbitrary submodules of $V$.
I have been trying to prove that submodules of free modules over von Neumann Regular rings must be finitely generated, which would give the result, but I am having no luck proving it. I am starting to think that this may be the wrong approach. I've also had no luck considering arbitrary exact sequences and showing that the short sequence of elements of the sequence tensored with $V$ is also exact. Any help would be much appreciated.