---
tag: 1R5P656GS6
kind: definition
name: Laurent polynomial ring
msc: [16S36, 13F20]
defines: QMY5JTZZB8
uses: [0B1261B5ED, 1HM7ZYMPD7]
sources:
  - url: https://en.wikipedia.org/wiki/Laurent_polynomial
---

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The *Laurent polynomial ring* $R[x, x^{-1}]$ is the [[monoid ring|1HM7ZYMPD7]] $R[\Z]$ of the additive group $\Z$, with the element $n \in \Z$ written $x^n$.
Its elements are the finite sums $\sum_{i = m}^{n} a_i x^i$ with $m \le n$ integers and $a_i \in R$.
Addition is termwise, and multiplication is determined by $x^i x^j = x^{i + j}$ for $i, j \in \Z$ and by $a x^i = x^i a$ for $a \in R$.

## 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|0GXMPDNYX3]].
The ring contains the [[polynomial ring|0SDC934R52]] $R[x]$, and $x$ is a [[unit|0XVEZDE9J8]] with inverse $x^{-1}$.
For a commutative ring $R$, $R[x, x^{-1}]$ is the [[localization|4C1AE84N5N]] of $R[x]$ at the multiplicative set $\set{1, x, x^2, \dots}$.
If $R$ is an [[integral domain|0QVD7WBFGN]], 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 $R[\Z^n]$.