Topological module explained

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

where 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

-adic topology on a ring and its modules. Let be an ideal of a ring The sets of the form for all and all positive integers form a base for a topology on that makes into a topological ring. Then for any left -module the sets of the form for all and all positive integers form a base for a topology on that makes into a topological module over the topological ring

References

• Book: Atiyah . Michael Francis . Michael Atiyah . MacDonald . I.G. . Ian G. Macdonald . Introduction to Commutative Algebra . Westview Press . 978-0-201-40751-8 . 1969.
• Book: Kuz'min, L. V.. Encyclopedia of Mathematics. Kluwer Academic Publishers. 1993. Hazewinkel. M.. 9. Topological modules.