# Topological module

*left topological module*

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

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

#### References

[1] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) |

[2] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |

**How to Cite This Entry:**

Topological module.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Topological_module&oldid=16407