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 + 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

• 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