Lemma 86.4.10. Let $B$ be a linearly topologized ring. The set of topologically nilpotent elements of $B$ is a closed, radical ideal of $B$. Let $\varphi : A \to B$ be a continuous map of linearly topologized rings.

If $f \in A$ is topologically nilpotent, then $\varphi (f)$ is topologically nilpotent.

If $I \subset A$ consists of topologically nilpotent elements, then the closure of $\varphi (I)B$ consists of topologically nilpotent elements.

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