Electric field along coaxial cylinders

Imagine two coaxial cylinders, one that is a volume with radius $Ra$ (and charge per unit lenght $-\lambda$) and another one that is just a surface with radius $Rc$ (and charge per unit lenght $+\lambda$), with $Rc > Ra$. What is the magnitude of the electric field everywhere in this distribution?

So the answer I have is this: $$E(r) = \begin{cases} 0, & \text{, if r < Ra} \\ \frac{1}{2\pi \mathcal{E}_0}\frac{\lambda}{r}, & \text{, if Ra<r<Rc} \\ 0, & \text{, if Rc < r} \end{cases}$$

My questions regard all the three positions. Why is the Electric field zero in the $r<Ra$ and the $Rc<r$ regions? And in the middle region, shouldn't the electric field be the sum of the fields created by the two charges?