I wonder if the following statement is true:
Every projective $R$-module is a direct sum of projective left ideals of $R$.
Most examples of non-free projective modules I have seen are all left ideals of $R$. Since a direct sum of projective modules is projective, I feel that it is reasonable to make a guess that every projective module may arise this way. My current belief is that the statement is false, but I haven't found a counterexample yet. (I also wonder if $R$'s being finitely generated would make a difference.) Please help.