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Topological module

## Summary

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 ${\displaystyle \mathbb {Z} ,}$  where ${\displaystyle \mathbb {Z} }$  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 ${\displaystyle I}$ -adic topology on a ring and its modules. Let ${\displaystyle I}$  be an ideal of a ring ${\displaystyle R.}$  The sets of the form ${\displaystyle x+I^{n}}$  for all ${\displaystyle x\in R}$  and all positive integers ${\displaystyle n,}$  form a base for a topology on ${\displaystyle R}$  that makes ${\displaystyle R}$  into a topological ring. Then for any left ${\displaystyle R}$ -module ${\displaystyle M,}$  the sets of the form ${\displaystyle x+I^{n}M,}$  for all ${\displaystyle x\in M}$  and all positive integers ${\displaystyle n,}$  form a base for a topology on ${\displaystyle M}$  that makes ${\displaystyle M}$  into a topological module over the topological ring ${\displaystyle R.}$