# Topological module

*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=41094