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" + "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

*springer|id=T/t093110|title=Topological ring|author=L. V. Kuzmin
*springer|id=T/t093060|title=Topological field|author=D. B. Shakhmatov
* Seth Warner: "Topological Rings". North-Holland, July 1993, ISBN 0444894462
* 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 0824793234.
* N. Bourbaki, "Éléments de Mathématique. Topologie Générale." Hermann, Paris 1971, ch. III §6