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.

General comments

The group of units of R may not be a topological group using the subspace topology, as inversion on the unit group need not be continuous with 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.) Embedding the unit group of R into the product R × R as (x,x−1) does make the unit group a topological group. (If inversion on the unit group is continuous in the subspace topology of R then the topology on the unit group viewed in R or in R × R as above 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 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 if and only if for every x in U there exists a natural number n such that x + In ⊆ 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