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.