Topological ring

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In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps

where R × R carries the product topology.

Contents

General comments

The group of units of R may not be a topological group using the subspace topology, as inversion on the unit group need not be continuous with 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.) Embedding the unit group of R into the product R × R as (x,x-1) does make the unit group a topological group. (If inversion on the unit group is continuous in the subspace topology of R then the topology on the unit group viewed in R or in R × R as above 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 which is a topological group (for +) in which multiplication is continuous, too.

Examples

Topological rings occur in mathematical analysis, for examples 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 algebra, the following construction is common: one starts with a commutative ring R containing an ideal I, and then considers the I-adic topology on R: 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 + InU. 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).

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