In
mathematics, a topological ring is a
ring that is also a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
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-s ...
. That means
is an additive
topological group
In mathematics, topological groups are logically 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 st ...
and a multiplicative
topological semigroup.
Topological rings are fundamentally related to
topological field
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
s 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 of a topological ring
is a
topological group
In mathematics, topological groups are logically 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 st ...
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 that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
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'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a reco ...
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
In mathematics, topological groups are logically 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 st ...
(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 (m ...
, for example as rings of continuous real-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s on some topological space (where the topology is given by pointwise convergence), or as rings of continuous
linear operators on some
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
; all
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s are topological rings. The
rational
Rationality is the quality of being guided by or based on reasons. 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 abi ...
,
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
,
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 numbers and
dual numbers
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
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 represen ...
for other low-dimensional examples.
In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, the following construction is common: one starts with 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. Most familiar as the name of ...
ring
containing an
ideal and then considers the
-adic topology on
: a
subset of
is open
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
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
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s is an example of an
-adic topology (with
).
Completion
Every topological ring is a
topological group
In mathematics, topological groups are logically 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 st ...
(with respect to addition) and hence a
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
subring such that the given topology on
equals the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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 (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
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 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 field
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
s. 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 fo ...
s and all its
subfields, and the
valued field
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
s, 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