Topological ring

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 $R\times R\to R$ where $R\times R$ 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.

General comments

The group of units Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R^{\times }} of a topological ring $R$ is a topological group when endowed with the topology coming from the embedding of $R^{\times }$ into the product $R\times R$ as $\left(x,x^{-1}\right).$ However, if the unit group is endowed with the subspace topology as a subspace of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R,} it may not be a topological group, because inversion on $R^{\times }$ 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 $R^{\times }$ is continuous in the subspace topology of $R$ then these two topologies on $R^{\times }$ 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.

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 linear operators on some normed vector space; all Banach algebras 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 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R,} the I-adic topology on $R$ is defined as follows: a subset $U$ of $R$ is open if and only if for every $x\in U$ there exists a natural number $n$ such that $x+I^{n}\subseteq U.$ 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 integers is an example of an $I$-adic topology (with $I=p\mathbb {Z}$).

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 $R$ is complete. If it is not, then it can be completed: one can find an essentially unique complete topological ring Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S} can be constructed as a set of equivalence classes of Cauchy sequences in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R,} this equivalence relation makes the ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c : R \to S} such that, for all CM Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : R \to T} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is Hausdorff and complete, there exists a unique CM Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g : S \to T} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = g \circ c.} If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \R \to \Q} endowed with the topology of pointwise convergence) the standard construction uses minimal Cauchy filters and satisfies the same universal property as above (see Bourbaki, General Topology, III.6.5).

The rings of formal power series and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -adic integers are most naturally defined as completions of certain topological rings carrying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} -adic topologies.

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 complex numbers and all its subfields, and the valued fields, which include the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -adic fields.

See also

• Compact group
• Complete field
• Locally compact field
• Locally compact group
• Ordered topological vector space
• Strongly continuous semigroup
• Topological abelian group
• Topological field
• Topological group – Group that is a topological space with continuous group operations
• Topological module
• Topological semigroup
• Topological vector space – Vector space with a notion of nearness

References

• Template:Springer
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• Warner, Seth (1989). Topological Fields. Elsevier. ISBN 9780080872681.
• Warner, Seth (1993). Topological Rings. 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, ISBN 0-8247-9323-4.
• N. Bourbaki, Éléments de Mathématique. Topologie Générale. Hermann, Paris 1971, ch. III §6