Editing Topological ring

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

Topological rings are fundamentally related to [[topological field]]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 (mathematics)|field]].{{sfn|Warner|1989|loc=Ch. II|p=77}}

==General comments==

The [[group of units]] <math>R^\times</math> of a topological ring <math>R</math> is a [[topological group]] when endowed with the topology coming from the [[Embedding#General topology|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]] 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]] of a [[global field]]; its unit group, called the [[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]] (for <math>+</math>) in which multiplication is continuous, too.

==Examples==

Topological rings occur in [[mathematical analysis]], for example as rings of continuous real-valued [[Function (mathematics)|function]]s on some topological space (where the topology is given by pointwise convergence), or as rings of continuous [[linear operator]]s on some [[normed vector space]]; all [[Banach algebra]]s are topological rings. The [[Rational number|rational]], [[Real number|real]], [[Complex number|complex]] and [[p-adic number|<math>p</math>-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 [[commutative algebra]], the following construction is common: given an [[Ideal (ring)|ideal]] <math>I</math> in a [[commutative]] ring <math>R,</math> the [[Adic topology|{{mvar|I}}-adic topology]] on <math>R</math> is defined as follows: a [[subset]] <math>U</math> of <math>R</math> is open [[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 space|Hausdorff]] if and only if the [[Intersection (set theory)|intersection]] of all powers of <math>I</math> is the zero ideal <math>(0).</math> 

The <math>p</math>-adic topology on the [[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]] (with respect to addition) and hence a [[uniform space]] in a natural manner. One can thus ask whether a given topological ring <math>R</math> is [[Complete uniform space|complete]]. 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 (topology)|dense]] [[subring]] such that the given topology on <math>R</math> equals the [[Subspace (topology)|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]]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 [[Nicolas Bourbaki|Bourbaki]], General Topology, III.6.5).

The rings of [[formal power series]] and the [[p-adic number|<math>p</math>-adic integers]] are most naturally defined as completions of certain topological rings carrying [[I-adic topology|<math>I</math>-adic topologies]].

==Topological fields==

Some of the most important examples are [[topological field]]s. A topological field is a topological ring that is also a [[Field (mathematics)|field]], and such that [[Multiplicative inverse|inversion]] of non zero elements is a continuous function. The most common examples are the [[complex number]]s and all its [[Subfield (mathematics)|subfields]], and the [[valued field]]s, which include the [[p-adic field|<math>p</math>-adic fields]].

==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]]|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]]|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]], ''Éléments de Mathématique. Topologie Générale.'' Hermann, Paris 1971, ch. III §6
{{refend}}

[[Category:Ring theory]]
[[Category:Topology]]
[[Category:Topological algebra]]
[[Category:Topological groups]]