How to show the ring $\Bbb Q[x,y,z]/(x^a,y^b,z^c)$ is noetherian? I know this ring is artin,and we can conclude the ring is artin because every ration ring is Noether.But I want to show this by definition of Noether ring.Thank you for your kind help.
2026-03-25 20:13:53.1774469633
The ring $\Bbb Q[x,y,z]/(x^a,y^b,z^c)$ is noetherian
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Hint: Every ideal of $R=\mathbb Q[x,y,z]/(x^a,y^b,z^c)$ is a linear subspace of $R$, viewed as a vector space over $\mathbb Q$. Now, observe that $$\dim_{\mathbb Q}R=abc < \infty$$