Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so.
In between step is $I^j/I^{j+1}$ is noetherian (artinian) $\forall j$. I am not getting this too.
Can anyone help?
2025-01-13 09:39:00.1736761140
Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so.
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Since $I^n=0$ the ideal $I^{n-1}$ is a finitely generated $R/I$-module, so it is noetherian (respectively, artinian).
Now $I^{n-2}/I^{n-1}$ is also a finitely generated $R/I$-module, so it is noetherian (respectively, artinian).
From the exact sequence $$0\to I^{n-1}\to I^{n-2}\to I^{n-2}/I^{n-1}\to 0$$ we get that $I^{n-2}$ is a noetherian (respectively, artinian) $R$-module.
Step by step, we get that $I^j$ is a noetherian (respectively, artinian) $R$-module for $j=n-1,n-2,\dots,1,0$, so $R$ is noetherian (respectively, artinian).