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

Answer 1

• $\Bbb Z[x, y]$ 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.