Topological module
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Examples
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]\displaystyle{ \Z, }[/math] where [math]\displaystyle{ \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]\displaystyle{ I }[/math]-adic topology on a ring and its modules. Let [math]\displaystyle{ I }[/math] be an ideal of a ring [math]\displaystyle{ R. }[/math] The sets of the form [math]\displaystyle{ x + I^n }[/math] for all [math]\displaystyle{ x \in R }[/math] and all positive integers [math]\displaystyle{ n, }[/math] form a base for a topology on [math]\displaystyle{ R }[/math] that makes [math]\displaystyle{ R }[/math] into a topological ring. Then for any left [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ M, }[/math] the sets of the form [math]\displaystyle{ x + I^n M, }[/math] for all [math]\displaystyle{ x \in M }[/math] and all positive integers [math]\displaystyle{ n, }[/math] form a base for a topology on [math]\displaystyle{ M }[/math] that makes [math]\displaystyle{ M }[/math] into a topological module over the topological ring [math]\displaystyle{ R. }[/math]
See also
- Linear topology
- Ordered topological vector space
- Topological abelian group
- Topological group – Group that is a topological space with continuous group action
- Topological ring
- Topological semigroup
- Topological vector space – Vector space with a notion of nearness
References
- Kuz'min, L. V. (1993). "Topological modules". in Hazewinkel, M.. Encyclopedia of Mathematics. 9. Kluwer Academic Publishers.
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