I have had trouble coming up with ideas to solving the following problem on pure modules. We have worked a couple of exercises from Jacobson and Hoffman and Kunze on pure modules but this is a sample test question and I do not see how to apply previous exercises to get the result.
A submodule $N$ of $M$ is pure if whenever $y \in N$ and $ a \in D$ are such taht there exists $x\in M$ with $ax = y$, then there exists $z \in N$ with $ az =y$
Question: Let $M$ be a finitely generated module over the polynomial ring $F[x]$, where $F$ is a field, and let $N$ be a pure submodule of $M$. Prove that there exists a submodule $L$ of $M$ such taht $N+L = M$ and $N \cap L = 0$
I was thinking that since all the hypothesis of http://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain apply to $M$ we can decompose $M$ into a pure and non pure part but I do not know how to proceed from that point.