Lemma 85.4.10. Let $\varphi : A \to B$ be a continuous map of linearly topologized rings.

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

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

Proof. Part (1) is clear. Let $g$ be an element of the closure of $\varphi (I)B$. Let $J \subset B$ be an open ideal. We have to show $g^ e \in J$ for some $e$. We have $g \in \varphi (I)B + J$ by Lemma 85.4.2. Hence $g = \sum _{i = 1, \ldots , n} f_ ib_ i + h$ for some $f_ i \in I$, $b_ i \in B$ and $h \in J$. Pick $e_ i$ such that $\varphi (f_ i^{e_ i}) \in J$. Then $g^{e_1 + \ldots + e_ n + 1} \in J$. $\square$

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