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When is the dual norm of an ideal $I$ of $R$ linearly generated by the norm of $I$?

Let $R$ be a commutative Noetherian with prime characteristic $p>0$ and $I$ be a nonzero two sided ideal of $R$. I was wondering when is the dual norm of $I$, which is by definition $N(I^*):=\{r\in R:rI^*\subseteq I^*\}$, linearly generated by $N(I^*)$? Namely, I am interested in the situation where $N(I^*)=N(I^*)^p$.

A:

You have the right idea, but you’re requiring that $N(I^*)=I$ because $rI^* \subseteq I^*$ is equivalent to $r \in I^*$.
Let $R$ be a local ring, and let $I$ be an ideal of $R$, and suppose that $I$ has a dual basis of length $n$ (e.g., $I$ is a vector space). Let $r \in R$. Then
$$
\bigl(r^{[p]}\bigr)\circ I^\ast = r^{[p]}\bigl(r^{ -1}I^\ast\bigr) \subseteq \bigl(r^{[p]}I^\ast\bigr) \subseteq I^\ast.
$$
For the last inclusion, we use the fact that $r^{[p]}I^\ast$ is generated by the set
$$
\bigl
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