Show that the following localization is Noetherian

Question

Let $R = \mathbb{Z}[x,y]/(xy-9)$. Consider the maximal ideal $(x, y, 3)$. Let $A$ be the localization of $R$ at $(x, y ,3)$.

I wish to show that this is Noetherian, but honestly, I don't really know where to start or what to consider.

Any insights or help is appreciated. Cheers

Is $\mathbb Z$ Noetherian? What about $\mathbb Z[x,y]$ then? — cqfd, Jan 07 '20 at 12:44

score 4 · Accepted Answer · answered Jan 07 '20 at 12:55

4

$\Bbb Z[x, y]$ is Noetherian
Any quotient of a (commutative) Noetherian ring is Noetherian
Any localisation of a Noetherian ring is Noetherian

You can try to show each of these three points. The second point follows directly from the definition of Noetherian, along with a suitable isomorphism theorem. The first point isn't too hard, while the third point is a bit more tricky, so I included a link.

answered Jan 07 '20 at 12:55

Arthur

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1

The first point follows the Hilbert basis theorem, right? Or is there a direct proof? – cqfd Jan 07 '20 at 13:29

Show that the following localization is Noetherian

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