Laurent polynomial ring

The ring of finite sums of a_i x^i allowing negative exponents.

Laurent polynomial ring

Let $R$ be a commutative ring with $1$ . The Laurent polynomial ring $R[x,x^{-1}]$ consists of all finite sums

\sum_{i=m}^n a_i x^i \quad (a_i\in R,\; m\le n,\; m\in\mathbb{Z}),

with the obvious addition and multiplication extending those of polynomial rings .

This is the result of adjoining an inverse to the indeterminate: $x$ becomes a unit in $R[x,x^{-1}]$ . Laurent polynomial rings are basic examples of localizations and appear naturally in algebraic geometry and representation theory.

Examples:

Over a field $k$ , $k[t,t^{-1}]$ is the coordinate ring of the multiplicative group $k^\times$ .
$\mathbb{Z}[q,q^{-1}]$ is the ring of Laurent polynomials in $q$ with integer coefficients.
The series $\sum_{n\ge 0} x^{-n}$ is not a Laurent polynomial (it has infinitely many negative-degree terms).