Ore extension

Summary

An Ore extension $R[x; \sigma, \delta]$ of a ring $R$ is a ring of polynomials $\sum a_i x^i$ in which the variable does not commute with the coefficients. Instead, $x a = \sigma(a) x + \delta(a)$ for an endomorphism $\sigma$ of $R$ and a twisted derivation $\delta$. Ore extensions give many of the basic examples of noncommutative rings, such as the Weyl algebras and quantum planes. They keep good properties of $R$, such as being a domain or being Noetherian, under mild conditions.