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.