Example of non-noetherian algebras which are tensor products of noetherian algebras

Question

We suppose all rings are commutative with unity.

I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian $A$-algebras.

The more examples the better. In other words, I'm asking a big list of examples.

Since you have asked similar questions and I have given answers, maybe you want to accept this or delete this question? ... — Martin Brandenburg, Feb 28 '14 at 21:36
I opened a meta thread on this. http://meta.math.stackexchange.com/questions/12899/why-was-this-question-asking-non-noetherian-examples-of-tensor-products-closed — Makoto Kato, Mar 02 '14 at 19:14

score 12 · Answer 1 · edited Apr 13 '17 at 12:21

$\mathbb{C} \otimes_{\mathbb{Q}} \mathbb{C}$ is not noetherian. For more examples from field extensions, see math.SE/19426 and your own question math.SE/694440. For example, if $K/F$ is a field extension which is not finitely generated, then $K \otimes_F K$ is not noetherian. From this one also gets more examples by localization, for example $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_q$ is not noetherian, see math.SE/684146. Likewise, if $k$ is a field, then $k[[x]] \otimes_k k[[y]]$ is not noetherian.

Example of non-noetherian algebras which are tensor products of noetherian algebras

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