In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a topological ring is a
ring that is also a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
such that both the addition and the multiplication are
continuous as maps:
where
carries the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. That means
is an additive
topological group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
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
The
group of units
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the ele ...
of a topological ring
is a
topological group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
when endowed with the topology coming from the
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
of
into the product
as
However, if the unit group is endowed with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative 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
In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global functio ...
; 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
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
(for
) in which multiplication is continuous, too.
Examples
Topological rings occur in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, 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 operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s on some
normed vector space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
; all
Banach algebras are topological rings. The
rational
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
,
real,
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
and
-adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane,
split-complex number
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...
s and
dual numbers form alternative topological rings. See
hypercomplex numbers
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represe ...
for other low-dimensional examples.
In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, the following construction is common: given an
ideal in a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
ring
the
-adic topology on
is defined as follows: a
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of
is open
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of all powers of
is the zero ideal
The
-adic topology on the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s is an example of an
-adic topology (with
).
Completion
Every topological ring is a
topological group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
(with respect to addition) and hence a
uniform space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
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
In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
such that the given topology on
equals the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
arising from
If the starting ring
is metric, the ring
can be constructed as a set of equivalence classes of
Cauchy sequence
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...
s 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
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
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 number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s and all its
subfields, and the
valued fields, which include the
-adic fields.
See also
*
*
*
*
*
*
*
*
*
*
*
*
Citations
References
*
*
*
*
* 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
{{refend
Ring theory
Topology
Topological algebra
Topological groups