In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Examples 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 Z , {\displaystyle \mathbb {Z} ,} where Z {\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 I {\displaystyle I} -adic topology on a ring and its modules. Let I {\displaystyle I} be an ideal of a ring R . {\displaystyle R.} The sets of the form x + I n {\displaystyle x+I^{n}} for all x ∈ R {\displaystyle x\in R} and all positive integers n , {\displaystyle n,} form a base for a topology on R {\displaystyle R} that makes R {\displaystyle R} into a topological ring. Then for any left R {\displaystyle R} -module M , {\displaystyle M,} the sets of the form x + I n M , {\displaystyle x+I^{n}M,} for all x ∈ M {\displaystyle x\in M} and all positive integers n , {\displaystyle n,} form a base for a topology on M {\displaystyle M} that makes M {\displaystyle M} into a topological module over the topological ring R . {\displaystyle R.}
See also
References
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.