Example of Noetherian ring over which the Euclidean algorithm is not valid.

Question

As stated in the question, I am looking for a Noetherian ring over which the Euclidean algorithm is not valid.

I am trying to construct non-trivial examples of Noetherian rings.

Thank you.

Answer 1

$R$ Noetherian implies $R[X]$ Noetherian.

In particular, $K[X,Y]=K[X][Y]$ is Noetherian. However $K[X,Y]$ is not a PID and so not Euclidean.

Also, $\mathbb Z[X]$ is Noetherian. However $\mathbb Z[X]$ is not a PID and so not Euclidean.

Here is a stronger example:

The ring of integers of $\mathbb{Q}(\sqrt{-19})$ is a PID, and hence Noetherian, but it is not Euclidean.