Quantum torus ($k_q[X^{\pm 1},Y^{\pm 1}]$)

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 $X^i{Y}^j$ 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.