Noetherian Rings - Dummit and Foote - Chapter 15 - exercise 10

[Math Amateur]

I am reading Dummit and Foote Chapter 15, Section 15.1: Noetherian Rings and Affine Algebraic Sets.

Exercise 10 reads as follows:

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Prove that the subring: $$k[x, x^2y, x^3y^2, ... ... ... \ , x^iy^{i-1} ... ... ]$$ of the polynomial ring k[x,y] is not a Noetherian ring and hence not a finitely generated k-algebra.

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Can someone please help me get a start on this exercise.

Peter[Note: This has also been posted on MHF]

 

[mathbalarka]

It's more or less obvious that the chain of ideals should be

$$(x) \subseteq (x, xy) \subseteq (x, xy, xy^2) \subseteq (x, xy, xy^2, xy^3) \subseteq \, \cdots$$

But the strict inclusions need to be settled. Define the ideals $I_n = (x, xy, xy^2, \cdots, xy^n)$ of $k[x, xy, xy^2, \cdots ]$. Clearly, $I_0 \not = I_1$, as $xy \notin I_0$. Furthermore, $I_2 \not = I_1$ as $xy^2$ can't be written as a $k$-linear combination of $x$ and $xy$.

Can you convince yourself in this way that $I_n \not = I_{n-1}$?

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Or you can just show (in the above approach) that the ideal $(x, xy, xy^2, \cdots)$ of $k[x, y]$ is not finitely generated, thus showing that $k[x, xy, xy^2, xy^3, \cdots ]$ is not finitely generated, which is equivalent to being non-Noetherian.