mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...

topological group can be considered as a topological module over

\Z,

where

\Z

is the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

with the discrete topology. A topological ring is a topological module over each of its

subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...

s. 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.

References

* Algebra Topology Topological algebra Topological groups {{algebra-stub

Examples

See also

References