Let $R$ be a Noetherian commutative unital ring. It is generally speaking not true that the tensor product of two Noetherian $R$-algebras is Noetherian (e.g. take $R$ to be a field, and consider the tensor square of some crazy field extension).

What is true is that a tensor product of a finite type $R$-algebra and a Noetherian $R$-algebra is Noetherian. It is not true, however, that an $R$-algebra whose tensor product with any Noetherian $R$-algebra is Noetherian has to be of finite type (consider the power series ring, for example). What is the name for the class of $R$-algebras that have this property?

localizationsof finitely generated algebras do. $\endgroup$ – Laurent Moret-Bailly May 1 '19 at 12:25