Topological ring

In mathematics, a topological ring is a ring, which is with respect to the addition of a topological group and the multiplication in said given topology is also continuous. If R is even a body, and is also the multiplicative inverses continuous, then it is called a topological field. Accordingly, one can define a topological division ring. In contrast to the non-commutative topological rings ( like the Endomorphismenringen below) are "real" topological division ring of little interest. Where not explicitly stated in this article on the body made ​​over statements also apply to skew field.

Local characterization of continuity

The continuity of the multiplication or the inverses can be in a ring R which is a topological group with respect to its addition, characterize alone with zero environments. To this end let B ( 0) is a basis of neighborhoods of 0: The left multiplication by a fixed element c is on R is continuous if

Accordingly, the continuity of the right multiplication by c can be characterized. In the case of a commutative ring, the two conditions are equivalent. If the left and right multiplication by any element c steadily and is still

Then the multiplication is continuous and R is a topological ring. The inverses is continuous in the invertible element, if for each U in B (0) a V exists in B (0 ), so that the inverses are of all. R is thus a field and this applies to all of its elements, then R is a topological body.

Properties. completion

• The conclusion of a sub- ring (or left ideal, right ideal, two-sided ideal ) is again a subring ( left ideal, right ideal, two-sided ideal ).
• In particular, the conclusion of the zero ideal is a two-sided ideal. The factor ring with the quotient topology is Hausdorff.
• Are available for each topological ring a substantially uniquely determined complete Hausdorff topological ring together with a steady ring homomorphism with kernel and dense range. is called a completion. In general, the completion of a topological body must not be a topological field more, but may even have a zero divisor.

Examples

Topological body

• The field of rational, real and complex numbers are topological body with respect to the usual topology ( the area defined by the absolute value function metric space ).
• More generally, are all valued field topological body. These include again the rational numbers with p- adic valuation (p prime). With respect to each p- adic valuation may result in a complete metric space are again completed a topological body, the body of the p- adic numbers.
• An example of a "real" topological skew field is the Quaternionenschiefkörper.

Endomorphism rings

• Important examples of topological rings provide the algebras A of continuous linear self-maps F a normed vector space V over a field K. As standard you put here the picture standard basis:

• These include the simplest examples, the full matrix rings of square matrices with entries in K. The standard can here be any norm on R instead of the standard figure, because all induce the same topology.

Note: The full endomorphism rings, apart from trivial cases not commutative, and no skew field. Often sub-rings are of interest, which occasionally have one of these properties:

• The ring D of the diagonal is a ( n> 1 real ) commutative bottom ring and hence of a topological ring.
• In general, all finite-dimensional algebras over a valued field can be represented as a matrix rings and so provided with a topology that is compatible with their links.

Function spaces

• Every Banach algebra. A particularly important example of this is C (T), the amount of continuous functions in a compact topological space T.

• The amount of holomorphic functions in a field is a topological ring ( even an integral domain ), the topology is the topology of compact convergence. On special areas in the complex plane unique representations of holomorphic functions there are possible:
• If G is the interior of a circular disc, then, each holomorphic function on G has a unique representation as a compact convergent power series. Conversely, the G on compact convergent power series is holomorphic on G.
• If G is a (right ) half-plane of the complex plane (ie G consists of all numbers z with Re ( z) > σ for a fixed real number σ ), then there exists a unique representation by a convergent Dirichlet series for G compact. Again, analogous convention applies to the power series to the reversal.