Topological module

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 existence of addition and scalar multiplication. ^[1]

A topological vector space is a topological module over a topological field.

An abelian topological group can be considered as a topological module over ${\displaystyle \mathbb {Z} ,}$ where ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle I}$-adic topology on a ring and its modules. Let ${\displaystyle I}$ be an ideal of a ring ${\displaystyle R.}$ The sets of the form ${\displaystyle x+I^{n}}$ for all ${\displaystyle x\in R}$ and all positive integers ${\displaystyle n,}$ form a base for a topology on ${\displaystyle R}$ that makes ${\displaystyle R}$ into a topological ring. Then for any left ${\displaystyle R}$-module ${\displaystyle M,}$ the sets of the form ${\displaystyle x+I^{n}M,}$ for all ${\displaystyle x\in M}$ and all positive integers ${\displaystyle n,}$ form a base for a topology on ${\displaystyle M}$ that makes ${\displaystyle M}$ into a topological module over the topological ring ${\displaystyle R.}$

• Atiyah, Michael Francis; MacDonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. ISBN 978-0-201-40751-8.
• Kuz'min, L. V. (1993). "Topological modules". In Hazewinkel, M. (ed.). Encyclopedia of Mathematics. Vol. 9. Kluwer Academic Publishers.