Laurent polynomial ring
= Laurent polynomial ring
{c}
{title2=$k[X_1^{\pm1},\ldots,X_r^{\pm1}]$}
A Laurent polynomial ring is the <localization of a ring> $k[X_1,\ldots,X_r]$ obtained by inverting all variables. Its elements are finite sums of <Laurent polynomials> with exponent vectors in $\mathbb Z^r$. It is a <Noetherian ring> by the <Hilbert basis theorem> and <Localization of a Noetherian ring>.