Laurent polynomial ring

Remarks

Example: $f = x^{-1} + 2$ and $g = x - 2x^2$ are Laurent polynomials, and $fg = 1 - 2x + 2x - 4x^2 = 1 - 4x^2$. Non-example: the series $\sum_{i \ge 0} x^i$ is not a Laurent polynomial, since it has infinitely many nonzero terms; it lies in the ring of Laurent series. The ring contains the polynomial ring $R[x]$, and $x$ is a unit with inverse $x^{-1}$. For a commutative ring $R$, $R[x, x^{-1}]$ is the localization of $R[x]$ at the multiplicative set $\set{1, x, x^2, \dots}$. If $R$ is an integral domain, the units of $R[x, x^{-1}]$ are the elements $u x^k$ with $u$ a unit of $R$ and $k \in \Z$. The ring in several variables $R[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ is the monoid ring $R[\Z^n]$.