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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nThe *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 $x^n$.\nIts elements are the finite sums $\\sum_{i = m}^{n} a_i x^i$ with $m \\le n$ integers and $a_i \\in R$.\nAddition 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 $a \\in R$.\n\n## Remarks\n\nExample: $f = x^{-1} + 2$ and $g = x - 2x^2$ are Laurent polynomials, and $fg = 1 - 2x + 2x - 4x^2 = 1 - 4x^2$.\nNon-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]].\nThe ring contains the [[polynomial ring|0SDC934R52]] $R[x]$, and $x$ is a [[unit|0XVEZDE9J8]] with inverse $x^{-1}$.\nFor a commutative ring $R$, $R[x, x^{-1}]$ is the [[localization|4C1AE84N5N]] of $R[x]$ at the multiplicative set $\\set{1, x, x^2, \\dots}$.\nIf $R$ 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 $k \\in \\Z$.\nThe ring in several variables $R[x_1^{\\pm 1}, \\dots, x_n^{\\pm 1}]$ is the monoid ring $R[\\Z^n]$.\n",
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