Topological ring explained
that is also a
topological space such that both the addition and the multiplication are continuous as maps:
where
carries the
product topology. That means
is an additive
topological group and a multiplicative
topological semigroup.
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.
General comments
of a topological ring
is a
topological group when endowed with the topology coming from the embedding of
into the product
as
However, if the unit group is endowed with the
subspace topology as a subspace of
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. If inversion on
is continuous in the subspace topology of
then these two topologies on
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 that 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
-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 a
commutative ring
the
-adic topology on
is defined as follows: a
subset
of
is open
if and only if for every
there exists a natural number
such that
This turns
into a topological ring. The
-adic topology is
Hausdorff if and only if the
intersection of all powers of
is the zero ideal
The
-adic topology on the
integers is an example of an
-adic topology (with
).
Completion
See main article: article and Completion (algebra).
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
is complete. If it is not, then it can be
completed: one can find an essentially unique complete topological ring
that contains
as a
dense subring such that the given topology on
equals the
subspace topology arising from
If the starting ring
is metric, the ring
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)
such that, for all CM
where
is Hausdorff and complete, there exists a unique CM
such that
If
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
-adic topologies.
Topological fields
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
-adic fields.
References
-
- Book: Warner, Seth. Topological Rings. Elsevier. 1993. 9780080872896.
- Vladimir I. Arnautov, Sergei T. Glavatsky and Aleksandr V. Michalev: Introduction to the Theory of Topological Rings and Modules. Marcel Dekker Inc, February 1996, .
- N. Bourbaki, Éléments de Mathématique. Topologie Générale. Hermann, Paris 1971, ch. III §6