Topological module

Summary

In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.

ExamplesEdit

A topological vector space is a topological module over a topological field.

An abelian topological group can be considered as a topological module over   where   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  -adic topology on a ring and its modules. Let   be an ideal of a ring   The sets of the form   for all   and all positive integers   form a base for a topology on   that makes   into a topological ring. Then for any left  -module   the sets of the form   for all   and all positive integers   form a base for a topology on   that makes   into a topological module over the topological ring  

See alsoEdit

ReferencesEdit

  • Kuz'min, L. V. (1993). "Topological modules". In Hazewinkel, M. (ed.). Encyclopedia of Mathematics. Vol. 9. Kluwer Academic Publishers.