# Topological module

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In mathematics, a **topological module** is a module over a topological ring such that scalar multiplication and addition are continuous.

## Examples[edit]

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*.