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If we change the ideal $$(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ to $$(X_1^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ in this problem, what is the answer to the raised question?

Again, the new local ring $R$ is of zero Krull dimension, and any ideal generated by a finite number of powers of $\bar X_i$'s is nilpotent.

Thanks for any cooperation!

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  • $\begingroup$ Btw, the change you made doesn't change much the question. The relations $x_i^{2^{i-1}}=0$ are replaced by $x_i^{2^{i}}=0$, so I expect the same answer. $\endgroup$
    – user26857
    May 2, 2016 at 8:10
  • $\begingroup$ @user26857 What is an example of an ideal which is neither f.g. nor idempotent, please? $\endgroup$
    – karparvar
    May 2, 2016 at 13:08

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