Topological ring

In mathematics , a topological ring is a ring which is a topological group in terms of addition and whose multiplication is also continuous in the given topology. If R is even a field and the multiplicative inverse formation is also continuous, then one speaks of a topological field . A topological oblique body can be defined accordingly . In contrast to the non-commutative topological rings (such as the endomorphism rings see below), “real” topological oblique bodies are of little interest. Where this is not expressly stated in this article, the statements made about bodies also apply to oblique bodies.

Local characterization of continuity

The continuity of the multiplication or the formation of the inverse can be characterized in a ring , which is a topological group with regard to its addition, only with zero environments . Let an environment basis of 0: The left multiplication with a fixed element is continuous if and only if ${\displaystyle R}$ ${\displaystyle B(0)}$
${\displaystyle c}$ ${\displaystyle R}$

for every environment in an environment in exists, so that holds. ${\displaystyle U}$ ${\displaystyle B(0)}$ ${\displaystyle V}$ ${\displaystyle B(0)}$ ${\displaystyle c\cdot V\subseteq U}$

Accordingly, the continuity of the law multiplication can also be characterized. In the case of a commutative ring, the two conditions are equivalent. Is the left and right multiplication continuous with each element and then still applies ${\displaystyle c}$ ${\displaystyle c}$

for all in there in so applies, ${\displaystyle U}$ ${\displaystyle B(0)}$ ${\displaystyle V}$ ${\displaystyle B(0)}$ ${\displaystyle V\cdot V\subseteq U}$

then the multiplication is continuous and a topological ring. The formation of the inverse is continuous in the invertible element if and only if an in exists for each in , so that the inverses of all lie in. So if there is a body and this is true for all of its elements , then it is a topological body. ${\displaystyle R}$
${\displaystyle x\in R^{\times }}$ ${\displaystyle U}$ ${\displaystyle B(0)}$ ${\displaystyle V}$ ${\displaystyle B(0)}$ ${\displaystyle x+V}$ ${\displaystyle x^{-1}+U}$ ${\displaystyle R}$ ${\displaystyle x\neq 0}$ ${\displaystyle R}$

Properties. completion

• The conclusion of a lower ring (or left ideal, right ideal, two-sided ideal) is again a lower ring (left ideal, right ideal, two-sided ideal).
• In particular, the conclusion of the null ideal is a two-sided ideal. The factor ring with the quotient topology is Hausdorffian . ${\displaystyle N}$ ${\displaystyle R/N}$
• For every topological ring there is an essentially uniquely determined complete Hausdorff topological ring together with a continuous ring homomorphism with a core and a dense image. is called the completion of . In general, however, the completion of a topological field no longer has to be a topological field, but can even have zero divisors. ${\displaystyle R}$ ${\displaystyle {\hat {R}}}$ ${\displaystyle R\to {\hat {R}}}$ ${\displaystyle N}$ ${\displaystyle {\hat {R}}}$ ${\displaystyle R}$

Examples

Topological bodies

• The fields of the rational, real and complex numbers are topological fields with respect to the usual topology (the metric space defined by the absolute value function ).
• More generally, all valued bodies are topological bodies. This again includes the rational numbers with an -adic valuation ( prime number). With regard to every -adic valuation , the field of -adic numbers can be completed to a complete metric space, again a topological field . ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle p}$
• An example of a “real” topological oblique body is the quaternion oblique body . ${\displaystyle \mathbb {H} }$

Endomorphism Rings

• Important examples of topological rings provide the algebras of continuous linear self-images of a normalized vector space over a field with . The standard is based on the mapping standard: ${\displaystyle A}$ ${\displaystyle F}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle \mathbb {Q} \subseteq K\subseteq \mathbb {C} }$

${\displaystyle ||F||_{A}:=\sup _{x\in V,\,||x||_{V}=1}\;||Fx||_{V}}$

• The simplest examples include the full die rings of the square matrices with entries . Instead of the mapping norm, the norm can be any norm since they all induce the same topology. ${\displaystyle R={\mathcal {M}}_{n\times n}(K)}$ ${\displaystyle K}$ ${\displaystyle R}$

Note: The full endomorphism rings are, apart from trivial cases, not commutative and also not oblique bodies. Often of interest are sub-rings that occasionally have one of these properties:

• The ring of the diagonal matrices is a (for real) commutative sub-ring of and thus a topological ring. ${\displaystyle D}$ ${\displaystyle n>1}$ ${\displaystyle {\mathcal {M}}_{n\times n}(K)}$
• In general, all finite-dimensional algebras can be represented as matrix rings over an evaluated body and thus provided with a topology that is compatible with their links.

Function rooms

See also : function space

complete topological rings in functional analysis:

• Any Banach algebra . A particularly important example is the set of continuous functions in a compact topological space . ${\displaystyle C(T)}$ ${\displaystyle T}$

topological rings in function theory:

• The set of holomorphic functions in a domain is a topological ring (even an integrity ring ), the topology is the topology of compact convergence . In special areas in the complex number level, clear representations of the holomorphic functions are possible: ${\displaystyle G\subseteq \mathbb {C} }$
  
• If the interior of a circular disk is, then each holomorphic function has a unique representation as a compact, convergent power series . Conversely, the power series converging on compact are holomorphic on . ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$
• If there is a (right) half-plane of the complex number plane (ie consists of all numbers with for a fixed real number ), then there is a unique representation by a Dirichlet series that converges to compact . The converse also applies here, analogously to the power series. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle z}$ ${\displaystyle Re(z)>\sigma }$ ${\displaystyle \sigma }$ ${\displaystyle G}$

literature

• Vladimir I. Arnautov, ST Glavatsky, Aleksandr V. Michalev: Introduction to the Theory of Topological Rings and Modules (= Pure and Applied Mathematics. Vol. 197) Marcel Dekker Inc, New York NY et al. 1996, ISBN 0-8247-9323- 4 .
• Nicolas Bourbaki : Eléments de mathématique. General topology. Hermann, Paris 1971, Section III § 6.
• Seth Warner : Topological Rings (= North-Holland Mathematics Studies. Vol. 178). North-Holland, Amsterdam et al. 1993, ISBN 0-444-89446-2 .

Any introductory textbook on these areas can be used for applications in functional analysis and function theory. See for example this literature on functional analysis and this on function theory.