TITLE: Strongly Noetherian property. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$?
QUESTION [9 upvotes]: 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 $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}$. 

When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian "commutative" $k$-algebras $A$ and $B$? 
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.

REPLY [4 votes]: Although the OP is mainly interested in noncommutative results and examples, let me say a few words about the commutative case. 
Let $k\subset K$ be a field extension. N. Bourbaki in Algebre. Chapitre 8,  Modules et anneaux semi-simples (edition 1958), exercise 22, page 99, gives the following criterion: if the extension $k\subset K$ is algebraic, then $K\otimes_kK$ is Noetherian iff $[K:k]<\infty$. 
Later on, in 1978, P. Vamos in the paper On the minimal prime ideals of a tensor product of two fields proves a more general result: for a field extension $k\subset K$, $K\otimes_kK$ is Noetherian iff $k\subset K$ is finitely generated. 
Inspired by Vamos' result, Resco, Small and Wadsworth in the paper Tensor products of division rings and finite generation of subfields prove a (partially) noncommutative result: let $D$ be a division algebra over a field $k$ and $k\subset K$ a commutative subfield of $D$. Then $D\otimes_kL$ is Noetherian iff the extension $k\subset K$ is finitely generated. As a by-product they get that $D\otimes_kD^0$ Noetherian implies $k\subset K$ finitely generated for every commutative subfield $K$ of $D$ containing $k$.