Topological Ring
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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 R 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: R \times R \to R where R \times R 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 R 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 ...
R^\times of a topological ring R 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 R^\times into the product R \times R as \left(x, x^\right). 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 R, it may not be a topological group, because inversion on R^\times 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 R^\times is continuous in the subspace topology of R then these two topologies on R^\times 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 p-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 I 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 R, the -adic topology on R 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 ...
U of R 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 x \in U there exists a natural number n such that x + I^n \subseteq U. This turns R into a topological ring. The I-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 I is the zero ideal (0). The p-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 I-adic topology (with I = p\Z).


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 R is complete. If it is not, then it can be ''completed'': one can find an essentially unique complete topological ring S that contains R 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 R 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 S. If the starting ring R is metric, the ring S 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 R, this equivalence relation makes the ring S Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) c : R \to S such that, for all CM f : R \to T where T is Hausdorff and complete, there exists a unique CM g : S \to T such that f = g \circ c. If R is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions f : \R \to \Q 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 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 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 p-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