Is this noncommutative ring Noetherian?

Question

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?

Answer 1

Trivially! A homomorphic image of a Noetherian ring is also Noetherian.

$A/I$ is left (or right) Noetherian if $A$ is left (or right) Noetherian for any ring $A$ and any ideal $I$.