This is a tutorial exercise for my Modules and Representation theory course.
Let $k$ be a field and $k\langle x,y \rangle$ denote the free algebra over $k$ with two generators. Let $q\in k^*$.
Is the ring $$k\langle x, y \rangle / (yx - qxy),$$ noetherian or artinian?
I know that $k\langle x, y \rangle$ itself is neither artinian nor noetherian. I also know that in the case of $q=1$, the ring is isomorphic to the polynomial ring and so is in that case it is noetherian (by Hilbert's basis theorem) but not artinian, since $(*)$ $I_n = (x^n)$ for $n\geq 0$ is a non stabilising descending chain.
However in the case where $q\neq 1$, I'm not sure how to proceed. Can I use the same chain $(*)$ to show that it's not artinian? And as for noetherian, I have no idea how to proceed.
Thanks in advance for any help.