Lemma 87.7.3. Let $A$ be a complete linearly topologized ring. Let $I \subset A$ be a finitely generated ideal such that the closure of $I^ n$ is open for all $n \geq 0$ and these closures form a fundamental system of open ideals. Then $A$ is adic and has a finitely generated ideal of definition.

**Proof.**
Denote $A'$ the ring $A$ endowed with the $I$-adic topology. The assumptions tells us that $A' \to A$ is taut. We conclude by Lemma 87.6.5 (to be sure, this lemma also tells us that $I$ is an ideal of definition).
$\square$

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