Topological module
≈ 1 min read
Names in other languages
No names in other languages are held for this entry yet.
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
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, 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.
See also
Text derived from https://en.wikipedia.org/w/index.php?oldid=1369225896 under CC BY-SA 4.0; modifications have been made.