As long as this is a draft, I am ignoring some style rules. I didn't sort the exports.jl so I don't lose track of what I've added

I have been playing around with some skew polynomial rings lately and so would also be interested in this functionality. I think this implementation here looks good. There are a few things I think would be worth discussing:

• Can we call the types OrePolyRing and OrePolyRingElem? This is just about the types. For the functions constructing the ring, we probably want aliases like ore_extension, skew_polynomial_ring or ore_polynomial_ring.
• We probably want to make ring parametrized by the type of the skew derivation:

@attributes mutable struct OrePolyRing{T<:RingElem, S, [U]}
  base_ring::Ring
  D::Symbol
  δ::S
  [sigma::U]

This has the advantage, that we can dispatch on the type S, for example, in the arithmetic. In my applications, $\sigma$ is always trivial and I think the implementation can be sped up in this case. In this regard, it might also be good idea to put the sigma in there (although it is a bit redundant).

• We probably want to support both R[x; sigma] and R[x; delta] (where in the first case delta = 0 is implied).

I really like the Sage interface: https://doc.sagemath.org/html/en/reference/noncommutative_polynomial_rings/sage/rings/polynomial/ore_polynomial_ring.html