Topological ring
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
where R × R carries the product topology
.
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 topological group. (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.)
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 which is a topological group
(for +) in which multiplication is continuous, too.
, for examples as rings of continuous real-valued function
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
; all Banach algebra
s 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 number
s 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 if and only if
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 integer
s is an example of an I-adic topology (with I = (p)).
(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.
s F. To have a topological field we should also specify that inversion
is continuous, when restricted to F\{0}. See the article on local field
s for some examples.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a topological ring is a ring R which is also a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
such that both the addition and the multiplication are continuous as maps
- R × R → R,
where R × R carries the product topology
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...
.
General comments
The group of units of R may not be a topological groupTopological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
using the subspace topology
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 natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
, as inversion on the unit group need not be continuous with the subspace topology. (An example of this situation is the adele ring
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....
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 topological group. (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.)
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 which is a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
(for +) in which multiplication is continuous, too.
Examples
Topological rings occur in mathematical analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, for examples as rings of continuous real-valued function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
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
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
; all Banach algebra
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 which at the same time is also a Banach space...
s are topological rings. The rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
, real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
and p-adic
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex number
Split-complex number
In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...
s and dual numbers form alternative topological rings. See hypercomplex numbers for other low dimensional examples.
In algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, 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
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
U of R is open if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
if and only if the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of all powers of I is the zero ideal (0).
The p-adic topology on the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is an example of an I-adic topology (with I = (p)).
Completion
Every topological ring is a topological groupTopological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
(with respect to addition) and hence a uniform space
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 which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
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
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
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
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
and the p-adic integers
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
are most naturally defined as completions of certain topological rings carrying I-adic topologies.
Topological fields
Some of the most important examples are also fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
s F. To have a topological field we should also specify that inversion
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
is continuous, when restricted to F\{0}. See the article on local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s for some examples.