Cylindrical coordinates of a ring

A classical problem for electromagnetism students is the calculation of the electric field on the central axis of a ring. It can be solved in many different ways, but I got stuck with the pure integration method. Let me introduce you the steps written in my book.

The electric field in an arbitrary point $mathbf{r}$ of the space, is given by the following expression:
$$
mathbf{E}(mathbf{r}) = dfrac{1}{4piepsilon_0} int_{L}{dfrac{lambda(mathbf{r’})(mathbf{r}-mathbf{r’})dl}{left| mathbf{r}-mathbf{r’} right|^3}}
$$

Then, in order to solve the integral, it says that in the central axis, the point in which we are calculating the field is given by $mathbf{r} = z,mathbf{u_z}$, which seems ok. However, the part where I get lost is regarding the position of each elemental charge $lambda(mathbf{r’})dl$, where the book says that this is $mathbf{r’} = R,mathbf{u_rho}$, where $R$ is the radius of the ring.

This also looks ok to me, except that not every differential charge is on the same $varphi$ cylindrical coordinate. So I don’t get why $mathbf{r’}$ has not the form $R,mathbf{u_rho} + varphi,mathbf{u_varphi}$.

And then, given the previous question, what would be the proper way of expressing a vector $(rho,varphi,z)$ as a sum of the basis and why?
$$
mathbf{v} = rho,mathbf{u_rho} + varphi,mathbf{u_varphi} + z,mathbf{u_z} quad text{or} quad mathbf{v} = rho,mathbf{u_rho} + rhovarphi,mathbf{u_varphi} + z,mathbf{u_z}
$$