The Stacks project

Lemma 87.5.8. Let $\varphi : A \to B$ be a continuous homomorphism of linearly topologized rings. If $\varphi $ is taut and $A$ is weakly pre-admissible, then $B$ is weakly pre-admissible.

Proof. Let $I \subset A$ be a weak ideal of definition. Then the closure $J$ of $IB$ is open and consists of topologically nilpotent elements by Lemma 87.4.10. Hence $J$ is a weak ideal of definition of $B$. $\square$

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