# Topological ring

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In [mathematics](/source/mathematics), a '''topological ring''' is a [ring](/source/Ring_(algebra)) <math>R</math> that is also a [topological space](/source/topological_space) such that both the addition and the multiplication are [continuous](/source/Continuity_(topology)) as maps{{sfn|Warner|1993|pp=1-2|loc=Def. 1.1}} <math>R \times R \to R</math> where <math>R \times R</math> carries the [product topology](/source/product_topology). This means <math>R</math> is an additive [topological group](/source/topological_group) and a multiplicative [topological semigroup](/source/topological_semigroup).

Topological rings are fundamentally related to [topological field](/source/topological_field)s and arise naturally while studying them, since for example the completion of a topological field may be a topological ring which is not a [field](/source/Field_(mathematics)).{{sfn|Warner|1989|loc=Ch. II|p=77}}

==General comments==

The [group of units](/source/group_of_units) <math>R^\times</math> of a topological ring <math>R</math> is a [topological group](/source/topological_group) when endowed with the topology coming from the [embedding](/source/Embedding) of <math>R^\times</math> into the product <math>R \times R</math> as <math>\left(x, x^{-1}\right).</math> However, if the unit group is endowed with the [subspace topology](/source/subspace_topology) as a subspace of <math>R,</math> it may not be a topological group, because inversion on <math>R^\times</math> need not be continuous with respect to the subspace topology. An example of this situation is the [adele ring](/source/adele_ring) of a [global field](/source/global_field); its unit group, called the [idele group](/source/idele_group), is not a topological group in the subspace topology. If inversion on <math>R^\times</math> is continuous in the subspace topology of <math>R</math> then these two topologies on <math>R^\times</math> 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](/source/topological_group) (with respect to addition) in which multiplication is continuous, too.

==Examples==

Topological rings occur in [mathematical analysis](/source/mathematical_analysis), for example as rings of continuous real-valued [function](/source/Function_(mathematics))s on some topological space (where the topology is given by pointwise convergence), or as rings of continuous [linear operator](/source/linear_operator)s on some [normed vector space](/source/normed_vector_space); all [Banach algebra](/source/Banach_algebra)s are topological rings. The [rational](/source/Rational_number), [real](/source/Real_number), [complex](/source/Complex_number) and [<math>p</math>-adic](/source/p-adic_number) numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, [split-complex number](/source/split-complex_number)s and [dual numbers](/source/dual_numbers) form alternative topological rings. See [hypercomplex numbers](/source/hypercomplex_numbers) for other low-dimensional examples.

In [commutative algebra](/source/commutative_algebra), the following construction is common: given an [ideal](/source/Ideal_(ring)) <math>I</math> in a [commutative](/source/commutative) ring <math>R,</math> the [{{mvar|I}}-adic topology](/source/Adic_topology) on <math>R</math> is defined as follows: a [subset](/source/subset) <math>U</math> of <math>R</math> is open [if and only if](/source/if_and_only_if) for every <math>x \in U</math> there exists a natural number <math>n</math> such that <math>x + I^n \subseteq U.</math> This turns <math>R</math> into a topological ring.  The <math>I</math>-adic topology is [Hausdorff](/source/Hausdorff_space) if and only if the [intersection](/source/Intersection_(set_theory)) of all powers of <math>I</math> is the zero ideal <math>(0).</math> 

The <math>p</math>-adic topology on the [integer](/source/integer)s is an example of an <math>I</math>-adic topology (with <math>I = p\Z</math>).

==Completion==
{{main article|Completion (algebra)}}

Every topological ring is a [topological group](/source/topological_group) (with respect to addition) and hence a [uniform space](/source/uniform_space) in a natural manner. One can thus ask whether a given topological ring <math>R</math> is [complete](/source/Complete_uniform_space). If it is not, then it can be ''completed'': one can find an essentially unique complete topological ring <math>S</math> that contains <math>R</math> as a [dense](/source/Dense_(topology)) [subring](/source/subring) such that the given topology on <math>R</math> equals the [subspace topology](/source/Subspace_(topology)) arising from <math>S.</math>
If the starting ring <math>R</math> is metric, the ring <math>S</math> can be constructed as a set of equivalence classes of [Cauchy sequence](/source/Cauchy_sequence)s in <math>R,</math> this equivalence relation makes the ring <math>S</math> Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) <math>c : R \to S</math> such that, for all CM <math>f : R \to T</math> where <math>T</math> is Hausdorff and complete, there exists a unique CM <math>g : S \to T</math> such that 
<math>f = g \circ c.</math> If <math>R</math> is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions <math>f : \R \to \Q</math> endowed with the topology of pointwise convergence) the standard construction uses minimal Cauchy filters and satisfies the same universal property as above (see [Bourbaki](/source/Nicolas_Bourbaki), General Topology, III.6.5).

The rings of [formal power series](/source/formal_power_series) and the [<math>p</math>-adic integers](/source/p-adic_number) are most naturally defined as completions of certain topological rings carrying [<math>I</math>-adic topologies](/source/I-adic_topology).

==Topological fields==

Some of the most important examples are [topological field](/source/topological_field)s. A topological field is a topological ring that is also a [field](/source/Field_(mathematics)), and such that [inversion](/source/Multiplicative_inverse) of non zero elements is a continuous function. The most common examples are the [complex number](/source/complex_number)s and all its [subfields](/source/Subfield_(mathematics)), and the [valued field](/source/valued_field)s, which include the [<math>p</math>-adic fields](/source/p-adic_field).

==See also==

* {{annotated link|Compact group}}
* {{annotated link|Complete field}}
* {{annotated link|Locally compact field}}
* {{annotated link|Locally compact group}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Strongly continuous semigroup}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological field}}
* {{annotated link|Topological group}}
* {{annotated link|Topological module}}
* {{annotated link|Topological semigroup}}
* {{annotated link|Topological vector space}}

==Citations==

{{reflist}}

==References==

{{refbegin}}
* {{springer|id=T/t093110|title=Topological ring|author=L. V. Kuzmin}}
* {{springer|id=T/t093060|title=Topological field|author=D. B. Shakhmatov}}
* {{cite book|last=Warner|first=Seth|title=Topological Fields|publisher=[Elsevier](/source/Elsevier)|url=https://www.elsevier.com/books/topological-fields/warner/978-0-444-87429-0|year=1989|isbn=9780080872681}}
* {{cite book|last=Warner|first=Seth|title=Topological Rings|publisher=[Elsevier](/source/Elsevier)|url=https://www.elsevier.com/books/topological-rings/warner/978-0-444-89446-5|year=1993|isbn=9780080872896}}
* 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|0-8247-9323-4}}.
* [N. Bourbaki](/source/N._Bourbaki), ''Éléments de Mathématique. Topologie Générale.'' Hermann, Paris 1971, ch. III §6
{{refend}}

Category:Ring theory
Category:Topological algebra
Category:Topological groups

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Adapted from the Wikipedia article [Topological ring](https://en.wikipedia.org/wiki/Topological_ring) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Topological_ring?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.