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Difference between revisions of "Topological module"

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''left topological module''
 
''left topological module''
  
An Abelian [[Topological group|topological group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930901.png" /> that is a [[Module|module]] over a [[Topological ring|topological ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930902.png" />, in which the multiplication mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930903.png" />, taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930904.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930905.png" />, is required to be continuous. A right topological module is defined analogously. Every submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930906.png" /> of a topological module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930907.png" /> is a topological module. If the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930908.png" /> is separated and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t0930909.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309011.png" /> is a separated module. A direct product of topological modules is a topological module. The completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309012.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309013.png" /> as an Abelian topological group can be given the natural structure of a topological module over the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309014.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309015.png" />.
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An [[Abelian group|Abelian]] [[topological group]] $A$ that is a [[Module|module]] over a [[topological ring]] $R$, in which the multiplication mapping $R \times A \to A$, taking $(r,a)$ to $ra$, is required to be continuous. A right topological module is defined analogously. Every submodule $B$ of a topological module $A$ is a topological module. If the module $A$ is separated and $B$ is closed in $A$, then $A/B$ is a separated module. A direct product of topological modules is a topological module. The completion $\hat A$ of the module $A$ as an Abelian topological group can be given the natural structure of a topological module over the completion $\hat R$ of the ring $R$.
  
A topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309017.png" />-module, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309018.png" /> is a topological group, is an Abelian topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309019.png" /> that is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309020.png" />-module, where the multiplication mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093090/t09309021.png" /> is required to be continuous.
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A topological $G$-module, where $G$ is a topological group, is an Abelian topological group $A$ that is a $G$-module, where the multiplication mapping $G \times A \to A$ is required to be continuous.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French) {{ZBL|0145.19302}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French) {{ZBL|0279.13001}}</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 18:09, 17 April 2017

left topological module

An Abelian topological group $A$ that is a module over a topological ring $R$, in which the multiplication mapping $R \times A \to A$, taking $(r,a)$ to $ra$, is required to be continuous. A right topological module is defined analogously. Every submodule $B$ of a topological module $A$ is a topological module. If the module $A$ is separated and $B$ is closed in $A$, then $A/B$ is a separated module. A direct product of topological modules is a topological module. The completion $\hat A$ of the module $A$ as an Abelian topological group can be given the natural structure of a topological module over the completion $\hat R$ of the ring $R$.

A topological $G$-module, where $G$ is a topological group, is an Abelian topological group $A$ that is a $G$-module, where the multiplication mapping $G \times A \to A$ is required to be continuous.

References

[1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) Zbl 0145.19302
[2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) Zbl 0279.13001
How to Cite This Entry:
Topological module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_module&oldid=16407
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article