Definition 15.35.1 (Topological rings). Let $R$ be a ring and let $M$ be an $R$-module.

We say $R$ is a

*topological ring*if $R$ is endowed with a topology such that both addition and multiplication are continuous as maps $R \times R \to R$ where $R \times R$ has the product topology. In this case we say $M$ is a*topological module*if $M$ is endowed with a topology such that addition $M \times M \to M$ and scalar multiplication $R \times M \to M$ are continuous.A

*homomorphism of topological modules*is just a continuous $R$-module map. A*homomorphism of topological rings*is a ring homomorphism which is continuous for the given topologies.We say $M$ is

*linearly topologized*if $0$ has a fundamental system of neighbourhoods consisting of submodules. We say $R$ is*linearly topologized*if $0$ has a fundamental system of neighbourhoods consisting of ideals.If $R$ is linearly topologized, we say that $I \subset R$ is an

*ideal of definition*if $I$ is open and if every neighbourhood of $0$ contains $I^ n$ for some $n$.If $R$ is linearly topologized, we say that $R$ is

*pre-admissible*if $R$ has an ideal of definition.If $R$ is linearly topologized, we say that $R$ is

*admissible*if it is pre-admissible and complete^{1}.If $R$ is linearly topologized, we say that $R$ is

*pre-adic*if there exists an ideal of definition $I$ such that $\{ I^ n\} _{n \geq 0}$ forms a fundamental system of neighbourhoods of $0$.If $R$ is linearly topologized, we say that $R$ is

*adic*if $R$ is pre-adic and complete.

Note that a (pre)adic topological ring is the same thing as a (pre)admissible topological ring which has an ideal of definition $I$ such that $I^ n$ is open for all $n \geq 1$.

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