# Topological ring

In mathematics, a **topological ring** is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps:^{[1]}
where carries the product topology. That means is an additive topological group and a multiplicative topological semigroup.

Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field.^{[2]}

## General comments

[edit]The group of units of a topological ring is a topological group when endowed with the topology coming from the embedding of into the product as However, if the unit group is endowed with the subspace topology as a subspace of it may not be a topological group, because inversion on need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on is continuous in the subspace topology of then these two topologies on are the same.

If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring that is a topological group (for ) in which multiplication is continuous, too.

## Examples

[edit]Topological rings occur in mathematical analysis, for example as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras are topological rings. The rational, real, complex and -adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples.

In commutative algebra, the following construction is common: given an ideal in a commutative ring the I-adic topology on is defined as follows: a subset of is open if and only if for every there exists a natural number such that This turns into a topological ring. The -adic topology is Hausdorff if and only if the intersection of all powers of is the zero ideal

The -adic topology on the integers is an example of an -adic topology (with ).

## Completion

[edit]Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ring is complete. If it is not, then it can be *completed*: one can find an essentially unique complete topological ring that contains as a dense subring such that the given topology on equals the subspace topology arising from
If the starting ring is metric, the ring can be constructed as a set of equivalence classes of Cauchy sequences in this equivalence relation makes the ring Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) such that, for all CM where is Hausdorff and complete, there exists a unique CM such that
If is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions endowed with the topology of pointwise convergence) the standard construction uses minimal Cauchy filters and satisfies the same universal property as above (see Bourbaki, General Topology, III.6.5).

The rings of formal power series and the -adic integers are most naturally defined as completions of certain topological rings carrying -adic topologies.

## Topological fields

[edit]Some of the most important examples are topological fields. A topological field is a topological ring that is also a field, and such that inversion of non zero elements is a continuous function. The most common examples are the complex numbers and all its subfields, and the valued fields, which include the -adic fields.

## See also

[edit]- Compact group – Topological group with compact topology
- Complete field – algebraic structure that is complete relative to a metric
- Locally compact field
- Locally compact quantum group – relatively new C*-algebraic approach toward quantum groups
- Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be defined
- Ordered topological vector space
- Strongly continuous semigroup – Generalization of the exponential function
- Topological abelian group – topological group whose group is abelian
- Topological field – Algebraic structure with addition, multiplication, and division
- Topological group – Group that is a topological space with continuous group action
- Topological module
- Topological semigroup – semigroup with continuous operation
- Topological vector space – Vector space with a notion of nearness

## Citations

[edit]**^**Warner 1993, pp. 1–2, Def. 1.1.**^**Warner 1989, p. 77, Ch. II.

## References

[edit]- L. V. Kuzmin (2001) [1994], "Topological ring",
*Encyclopedia of Mathematics*, EMS Press - D. B. Shakhmatov (2001) [1994], "Topological field",
*Encyclopedia of Mathematics*, EMS Press - Warner, Seth (1989).
*Topological Fields*. Elsevier. ISBN 9780080872681. - Warner, Seth (1993).
*Topological Rings*. Elsevier. ISBN 9780080872896. - Vladimir I. Arnautov, Sergei T. Glavatsky and Aleksandr V. Michalev:
*Introduction to the Theory of Topological Rings and Modules*. Marcel Dekker Inc, February 1996, ISBN 0-8247-9323-4. - N. Bourbaki,
*Éléments de Mathématique. Topologie Générale.*Hermann, Paris 1971, ch. III §6