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Topological module

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated August 13, 2026

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

01Examples

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 the 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 \mathbb {Z} , where \mathbb {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.

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This article is adapted from the Wikipedia article “Topological module”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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