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
- R × R → R,
where R × R carries the product topology. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural
Contents |
General comments
The group of units of R may not be a topological group using the subspace topology, as inversion on the unit group need not be continuous with the subspace topology. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is (An example of this situation is the adele ring of a global field. In Number theory, the adele ring is a Topological ring which is built on the field of Rational numbers (or more generally any Algebraic Its unit group, called the idele group, is not a topological group in the subspace topology. In Mathematics, an adelic algebraic group is a Topological group defined by an Algebraic group G over a Number field K ) 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. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the
Examples
Topological rings occur in mathematical analysis, for examples 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. Analysis has its beginnings in the rigorous formulation of Calculus. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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 In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the The rational, real, complex and p-adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In the plane, split-complex numbers and dual numbers form alternative topological rings. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real A variety of dualities in mathematics are listed at Duality (mathematics. See hypercomplex numbers for other low dimensional examples. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic
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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, commutativity is the ability to change the order of something without changing the end result In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. ↔ 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). In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently
The p-adic topology on the integers is an example of an I-adic topology (with I = (p)). The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French
Completion
Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. In Commutative algebra, the term completion refers to several related Functors on Topological rings and modules In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In the Mathematical field of Topology, a uniform space is a set with a uniform structure. One can thus ask whether a given topological ring R is complete. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has 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. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is The ring S can be constructed as a set of equivalence classes of Cauchy sequences in R. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence
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. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897
Topological fields
Some of the most important examples are also fields F. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division To have a topological field we should also specify that inversion is continuous, when restricted to F\{0}. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which See the article on local fields for some examples. 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
References
- L. V. Kuzmin (2001), “Topological ring”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- D. The Encyclopaedia of Mathematics is a large reference work in Mathematics. B. Shakhmatov (2001), “Topological field”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Seth Warner: Topological Rings. The Encyclopaedia of Mathematics is a large reference work in Mathematics. North-Holland, July 1993, ISBN 0444894462
- Vladimir I. Arnautov, Sergei T. Glavatsky and Aleksandr V. Michalev: Introduction to the Theory of Topological Rings and Modules. Marcel Dekker Inc, February 1996, ISBN 0824793234.
- N. Bourbaki, Éléments de Mathématique. Topologie Générale. Hermann, Paris 1971, ch. III §6
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic