Topological ring

 ( Ring Theory ) 

In mathematics, a topological ring is a ring $R$ that is also a topological space such that both the addition and the multiplication are continuous as maps $R\; \backslash times\; R\; \backslash to\; R$ where $R\; \backslash times\; R$ carries the product topology. This means $R$ is an additive topological group and a multiplicative topological semigroup.

Topological rings are fundamentally related to topological fields 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.

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General comments The group of units $R^\backslash times$ of a topological ring $R$ is a topological group when endowed with the topology coming from the embedding of $R^\backslash times$ into the product $R\; \backslash times\; R$ as $\backslash left(x,\; x^\{-1\}\backslash right).$ However, if the unit group is endowed with the subspace topology as a subspace of $R,$ it may not be a topological group, because inversion on $R^\backslash times$ 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 $R^\backslash times$ is continuous in the subspace topology of $R$ then these two topologies on $R^\backslash times$ 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 (with respect to addition) in which multiplication is continuous, too.

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Examples Topological rings occur in mathematical analysis, 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 on some normed vector space; all are topological rings. The Rational number, Real number, Complex number and p-adic number numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex numbers 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 $I$ in a commutative ring $R,$ the Adic topology on $R$ is defined as follows: a subset $U$ of $R$ is open if and only if for every $x\; \backslash in\; U$ there exists a natural number $n$ such that $x\; +\; I^n\; \backslash subseteq\; U.$ This turns $R$ into a topological ring. The $I$-adic topology is Hausdorff space if and only if the intersection of all powers of $I$ is the zero ideal $(0).$

The $p$-adic topology on the is an example of an $I$-adic topology (with $I\; =\; p\backslash Z$).

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Completion 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 $R$ 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 such that the given topology on $R$ equals the subspace topology arising from $S.$ If the starting ring $R$ is metric, the ring $S$ can be constructed as a set of equivalence classes of 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) $c\; :\; R\; \backslash to\; S$ such that, for all CM $f\; :\; R\; \backslash to\; T$ where $T$ is Hausdorff and complete, there exists a unique CM $g\; :\; S\; \backslash to\; T$ such that $f\; =\; g\; \backslash circ\; c.$ If $R$ is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions $f\; :\; \backslash R\; \backslash to\; \backslash Q$ 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, General Topology, III.6.5).

The rings of formal power series and the p-adic number are most naturally defined as completions of certain topological rings carrying I-adic topology.

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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 and all its subfields, and the , which include the p-adic field.

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See also

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Citations

• Seth Warner (1989). 9780080872681, Elsevier. . ISBN 9780080872681
• Seth Warner (1993). 9780080872896, Elsevier. . 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, .
• N. Bourbaki, Éléments de Mathématique. Topologie Générale. Hermann, Paris 1971, ch. III §6

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