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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?

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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 $I$.

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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
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