Could be proved the cyclic vector theorem for a differential module (over a differential field of characteristic zero) by using the fact that the ring of differential operators (over the same differential field) is simple? If yes, Is such a proof known? Let me also ask: Are these two statements (i.e cyclic vector theorem and simplicity of the ring of differential operators) equivalent in this context?
P.S. I am only considering the one variable case.
Thanks in advance for your cooperation