Lemma 15.35.2. Let $\varphi : R \to S$ be a ring map. Let $I \subset R$ and $J \subset S$ be ideals and endow $R$ with the $I$-adic topology and $S$ with the $J$-adic topology. Then $\varphi $ is a homomorphism of topological rings if and only if $\varphi (I^ n) \subset J$ for some $n \geq 1$.

**Proof.**
Omitted.
$\square$

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