Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D.
Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is free, is $R$ a PID?
I couldn't find my second question here, only the first one.
Another question regarding this if it's true is how much can we "stretch" the conditions, this is, could be concluded that $R$ is a PID knowing that some submodules of a free $R$-module verify some condition? "Some condition" may sounds vague, but I couldn't think of any that will do.