Flipped non-associative polynomial rings and the Cayley–Dickson construction

We introduce and study flipped non-associative polynomial rings. In particular, we show that all Cayley–Dickson algebras naturally appear as quotients of a certain type of such rings; this extends the classical construction of the complex numbers (and quaternions) as a quotient of a (skew) polynomial ring to the octonions, and beyond. We also extend some classical results on algebraic properties of Cayley–Dickson algebras by McCrimmon to a class of flipped non-associative polynomial rings.