Lemma 23.4.4. Let $(A, I, \gamma )$ be a divided power ring. Let $E \subset I$ be a subset. Then the smallest ideal $J \subset I$ preserved by $\gamma $ and containing all $f \in E$ is the ideal $J$ generated by $\gamma _ n(f)$, $n \geq 1$, $f \in E$.

**Proof.**
Follows immediately from Lemma 23.4.3.
$\square$

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