View source for Topological module

In [[Mathematics|mathematics]], a '''topological module''' is a module over a [[Topological ring|topological ring]] such that [[Scalar multiplication|scalar multiplication]] and addition are continuous.
==Examples==

A [[Topological vector space|topological vector space]] is a topological module over a topological field.

An [[Abelian group|abelian]] [[Topological group|topological group]] can be considered as a topological module over <math>\Z,</math> where <math>\Z</math> is the [[Ring of integers|ring of integers]] with the discrete topology.

A topological ring is a topological module over each of its [[Subring|subring]]s.

A more complicated example is the <math>I</math>-adic topology on a ring and its modules. Let <math>I</math> be an [[Ideal (ring theory)|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 (topology)|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==

* {{annotated link|Linear topology}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological group}}
* {{annotated link|Topological ring}}
* {{annotated link|Topological semigroup}}
* {{annotated link|Topological vector space}}

==References==

* {{Cite book|last=Kuz'min|first=L. V.|title=Encyclopedia of Mathematics|publisher=Kluwer Academic Publishers|year=1993|editor-last=Hazewinkel|editor-first=M.|volume=9|chapter=Topological modules}}

[[Category:Algebra]]
[[Category:Topology]]
[[Category:Topological algebra]]
[[Category:Topological groups]]


{{Sourceattribution|Topological module}}