Let $k$ be a field. It is well-known that $A\otimes\_{k}B$ is not necessarily Noetherian even if $k$-algebras $A$ and $B$ are Noetherian. For example $\mathbb{R}\otimes\_{\mathbb{Q}}\mathbb{R}$. 


1. When is the tensor $A\otimes\_{k}B$ Noetherian for Noetherian "commutative" $k$-algebras $A$ and $B$?
2. What if $A$ is noncommutative? Are there any good criteria for $A\otimes\_{k}B$ to be Noetherian? If $B$ is a finitely generated $k$-algebra, Hilbert's basis theorem implies that $A\otimes\_{k}B$ is again Noetherian. So we need to check this with quite nasty $B$.


My primary motivation to ask these questions is the second question. Such ring $A$ is called a "strongly Noethrian ring" and has a lot of good properties, but I don't know many examples. Moreover I realized that things are not very clear even in commutative case and I need to understand commutative case first. I would appreciate it if experts on MO could let me know good criteria for this property and provide me with examples. 


Rings I have in my mind are weakly noncommutative in the sense that they are commutative up to scalar multiplication such as quantum planes and their $good$ hypersurfaces. 



Strongly Noetherian property. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$?
You could try having a look at Yekutieli and Zhang's paper Homological Transcendence Degree (<http://arxiv.org/abs/math/04120130>). They call a $k$-algebra $A$ "doubly Noetherian" if $A \otimes\_k A^{op}$ is Noetherian, and "rationally Noetherian" if $A \otimes\_k U$ is Noetherian for every division ring $U$.


There's a lot of other stuff in the paper too, but the first couple of sections look at several conditions for when rings (usually simple Artinian or division) are doubly or rationally noetherian, and there's a nice example of a field in section 7 which is not doubly noetherian.


It's not quite what you asked (it certainly won't help with the $q$-plane, but maybe the $q$-torus or the $q$-division ring?) but it might help with intuition and provide some examples.


