Topological ring

that is also a topological space such that both the addition and the multiplication are continuous as maps:[1]

carries the product topology.

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.

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.

is continuous in the subspace topology of

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

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

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

there exists a natural number

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

-adic topology on the integers is an example of an

Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.

as a dense subring such that the given topology on

equals the subspace topology arising from

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)

is Hausdorff and complete, there exists a unique CM

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

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