In mathematics, a '''topological module''' is a module over a topological ring such that scalar multiplication and addition are continuous.
==Examples== A module topology is the finest topology such that scalar multiplication and addition are continuous. A finitely generated module topology is a 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 is a topological module over a topological field.
An abelian topological group can be considered as a topological module over <math>\Z,</math> where <math>\Z</math> is the ring of integers with the discrete topology.
A topological ring is a topological module over each of its subrings.
A more complicated example is the <math>I</math>-adic topology on a ring and its modules. Let <math>I</math> be an ideal 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 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 | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}} * {{Cite book|last=Kuz'min|first=L. V.|title=Encyclopedia of Mathematics|publisher=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
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