Quantum torus
= Quantum torus
{title2=$k_q[X^{\pm1},Y^{\pm1}]$}
The two-dimensional quantum torus is the <algebra> with invertible generators $X,Y$ and relation $YX=qXY$, for $q\in k^\times$. Its <basis> consists of $X^iY^j$ with $i,j\in\mathbb Z$, with the same multiplication rule as the <quantum plane>. It is a <right Noetherian ring>: successive adjunction of $X^{-1}$, $Y$, and $Y^{-1}$ satisfies the hypotheses of the <noncommutative Hilbert basis theorem>.