# 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.

### Examples

Topological rings occur in mathematical analysis, for examples 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 iff 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}.