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 , 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 , 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".