Algebra, Problems For Fun (21)

Let $\displaystyle K$ be a field, $\displaystyle \sigma \in \text{Aut}(K),$ and $\displaystyle x$ an indeterminate. Let $\displaystyle R=\{a_nx^n + a_{n-1}x^{n-1} + \cdots + a_mx^m : \ a_j \in K, \ n \in \mathbb{Z} \}.$ Define addition and multiplication in $\displaystyle R$ exactly the same as what we have

in the ordinary polynomials __but__ with this extra rule that $\displaystyle xa=\sigma(a) x,$ for all $\displaystyle a \in K.$ So, for example $\displaystyle (x^2 -2x + x^{-1})(3x)=\sigma^2(3)x^3-2\sigma(3)x^2+\sigma^{-1}(3).$

It's easy to see that $\displaystyle R$ is a ring with identity and that $\displaystyle R$ is commutative if and only if $\displaystyle \sigma=\text{id}_K.$ The ring $\displaystyle R$ is called the ring of Laurent polynomials with the twist $\displaystyle \sigma.$ A standard notation for $\displaystyle R$ is

$\displaystyle K[x,x^{-1}, \sigma]$ or $\displaystyle K[x^{\pm 1}, \sigma].$ These rings are very important in ring theory and noncommutative algebraic geometry.

__Problem__: Find the center of $\displaystyle R.$

__Warning__: