In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Examples
A module topology is the finest topology such that scalar multiplication and addition are continuous. A finitely generated module topology is a topological ring. Note that this general definition of a module topology does not need to have a ring structure, it merely needs existence of addition and scalar multiplication.
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>\Z,</math> where <math>\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>I</math>-adic topology on a ring and its modules. Let <math>I</math> be an ideal of a ring <math>R.</math> The sets of the form <math>x + I^n</math> for all <math>x \in R</math> and all positive integers <math>n,</math> form a base for a topology on <math>R</math> that makes <math>R</math> into a topological ring. Then for any left <math>R</math>-module <math>M,</math> the sets of the form <math>x + I^n M,</math> for all <math>x \in M</math> and all positive integers <math>n,</math> form a base for a topology on <math>M</math> that makes <math>M</math> into a topological module over the topological ring <math>R.</math>