# Topological module

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Topological_module
> Markdown URL: https://mediated.wiki/source/Topological_module.md
> Source: https://en.wikipedia.org/wiki/Topological_module
> Source revision: 1351899342
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/mathematics), a '''topological module''' is a [module](/source/Module_(algebra)) over a [topological ring](/source/topological_ring) such that [scalar multiplication](/source/scalar_multiplication) and addition are [continuous](/source/Continuous_function_(topology)).

==Examples==
A module topology is the finest topology such that [scalar multiplication](/source/scalar_multiplication) and addition are [continuous](/source/Continuous_function_(topology)). A finitely generated module topology is a [topological ring](/source/topological_ring). Note that this general definition of a module topology does not need to have a ring structure, it merely needs existence of addition and scalar multiplication. <ref>https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ModuleTopology.html#IsModuleTopology.isTopologicalRing</ref>

A [topological vector space](/source/topological_vector_space) is a topological module over a [topological field](/source/topological_field).

An [abelian](/source/abelian_group) [topological group](/source/topological_group) can be considered as a topological module over <math>\Z,</math> where <math>\Z</math> is the [ring of integers](/source/ring_of_integers) with the [discrete topology](/source/discrete_topology).

A topological ring is a topological module over each of its [subring](/source/subring)s.

A more complicated example is the <math>I</math>-[adic topology](/source/adic_topology) on a ring and its modules. Let <math>I</math> be an [ideal](/source/Ideal_(ring_theory)) of a ring <math>R.</math> The sets of the form <math>x + I^n</math> for all <math>x \in R</math> and all positive integers <math>n,</math> form a [base](/source/Base_(topology)) for a topology on <math>R</math> that makes <math>R</math> into a topological ring. Then for any left <math>R</math>-module <math>M,</math> the sets of the form <math>x + I^n M,</math> for all <math>x \in M</math> and all positive integers <math>n,</math> form a base for a topology on <math>M</math> that makes <math>M</math> into a topological module over the topological ring <math>R.</math>

==See also==
{{div col}}
* {{annotated link|Linear topology}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological field}}
* {{annotated link|Topological group}}
* {{annotated link|Topological ring}}
* {{annotated link|Topological semigroup}}
* {{annotated link|Topological vector space}}
{{div col end}}

== References ==
{{Reflist}}
*{{Cite book | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=MacDonald | first2=I.G. | author2-link=Ian G. Macdonald | title=[Introduction to Commutative Algebra](/source/Introduction_to_Commutative_Algebra) | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}}
* {{Cite book|last=Kuz'min|first=L. V.|title=[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)|publisher=[Kluwer Academic Publishers](/source/Kluwer_Academic_Publishers)|year=1993|editor-last=Hazewinkel|editor-first=M.|editor-link=Michiel Hazewinkel|volume=9|chapter=Topological modules}}

Category:Abstract algebra
Category:Topology
Category:Topological algebra
Category:Topological groups

{{topology-stub}}
{{algebra-stub}}

---
Adapted from the Wikipedia article [Topological module](https://en.wikipedia.org/wiki/Topological_module) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Topological_module?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.