Topological ring
From Wikipedia, the free encyclopedia
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. It is false that the unit group of R is a topological group in general 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 toplogical group. (As an exercise, one can check that 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.)
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[edit] 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 + 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)).
[edit] 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.
[edit] 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}.
