# Is this noncommutative ring Noetherian?

Let $A$ be a quantum $\mathbb{P}^n$ defined by $$A=\mathbb{C}\langle x_1,\dots, x_{n+1}\rangle/(x_ix_j-r_{ij} x_j x_i)_{i,j}.$$ This is known to be Noetherian. Given a homogeneous polynomial $f$ in $x_i,\dots x_{n+1}$ and assume $f$ lies in the center of $A$. Is $A/(f)$ Noetherian ring?

-

$A/I$ is left (or right) Noetherian if $A$ is left (or right) Noetherian for any ring $A$ and any ideal $I$.
You are right. I totally forgot the fact. I was too scared with noncommutative rings. It doesn't matter whether $f$ is in the center or not. –  M. K. Oct 2 '12 at 18:44