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Assuming we are in a Noetherian ring $R$ where such an integer always exists for a given ideal $I$, I am looking for any information on properties this integer possesses (in relation to $I$).

Formally, let $n : = \text{min} \{k \in \mathbb{N} \mid \sqrt{I}^k \subseteq{I} \}$. That $n$ exists when $R$ is Noetherian is Proposition 7.14 in Atiyah-McDonald.

For example, some obvious properties include:

  • $n = 0 \iff I = R$.
  • $n = 1 \iff I = \sqrt{I}$ and $I$ is proper.

It could be the case that we only achieve certain properties when we assume $R$ is a local ring. It is OK to make that assumption.

Thanks for your consideration. Let me know if this question is too open-ended.

beanshadow
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  • It might be easier to think about it once we quotient by I. In that case, the smallest such n is measuring how much nilpotence there is in some sense, at least in the case of a principal ideal. – Asvin Jun 29 '20 at 18:32
  • You may want to google "effective Nullstellensatz" and start from there. – Youngsu Jul 01 '20 at 04:02

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