Skew polynomial ring

Remarks

The product of monomials is $(a x^i)(b x^j) = a \sigma^i(b) x^{i+j}$. Example: let $R = \C$ and let $\sigma$ be complex conjugation. Then $x z = \bar z x$ for $z \in \C$, and $x^2 z = x \bar z x = z x^2$, so $x^2$ commutes with all of $\C$ and with $x$; it lies in the center. Example: for a field $k$, $q \in k$ nonzero, $R = k[y]$ and $\sigma(y) = q y$, the ring $R[x; \sigma]$ has $x y = q y x$; it is the quantum plane. Non-example: the first Weyl algebra, with $x y = y x + 1$, is an Ore extension $k[y][x; \id, d/dy]$ with a nonzero derivation, so it is not of the form $R[x; \sigma]$ with this $R$ and this $x$. When $\sigma = \id$ the ring $R[x; \sigma]$ is the ordinary polynomial ring $R[x]$. Some books call every Ore extension $R[x; \sigma, \delta]$ a skew polynomial ring.