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