[Sections 7.1 and 7.2, EGA1]

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

1. 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.

2. 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.

3. 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.

4. 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 $n$.

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

6. If $R$ is linearly topologized, we say that $R$ is admissible if it is pre-admissible and complete1.

7. 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 $0$.

8. 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 $n \geq 1$.

[1] By our conventions this includes separated.

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