# Topological ring

In mathematics, a **topological ring** is a ring *R* which is also a topological space such that both the addition and the multiplication are continuous as maps

*R*×*R*→*R*,

where *R* × *R* carries the product topology. That means *R* is an additive topological group and a multiplicative topological semigroup.

## General comments

The group of units *R*^{×} of *R* is a topological group when endowed with the topology coming from the embedding of *R*^{×} into the product *R* × *R* as (*x*,*x*^{−1}). However, if the unit group is endowed with the subspace topology as a subspace of *R*, it may not be a topological group, because inversion on *R*^{×} 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 *R*^{×} is continuous in the subspace topology of *R* then these two topologies on *R*^{×} 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 which is a topological group (for +) in which multiplication is continuous, too.

## Examples

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 *p*-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 algebra, the following construction is common: one starts with a commutative ring *R* containing an ideal *I*, and then considers the ** I-adic topology** on

*R*: a subset

*U*of

*R*is open if and only if for every

*x*in

*U*there exists a natural number

*n*such that

*x*+

*I*

^{n}⊆

*U*. This turns

*R*into a topological ring. The

*I*-adic topology is Hausdorff if and only if the intersection of all powers of

*I*is the zero ideal (0).

The *p*-adic topology on the integers is an example of an *I*-adic topology (with *I* = (*p*)).

## Completion

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 *R* is complete. If it is not, then it can be *completed*: one can find an essentially unique complete topological ring *S* which contains *R* as a dense subring such that the given topology on *R* equals the subspace topology arising from *S*.
The ring *S* can be constructed as a set of equivalence classes of Cauchy sequences in *R*.

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

## Topological fields

Some of the most important examples are also fields *F*. To have a **topological field** we should also specify that inversion is continuous, when restricted to *F*\{0}. See the article on local fields for some examples.

## References

- L. V. Kuzmin (2001), "Topological ring", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - D. B. Shakhmatov (2001), "Topological field", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Seth Warner:
*Topological Rings*. North-Holland, July 1993, ISBN 0-444-89446-2 - 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