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.
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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 realvalued 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 padic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, splitcomplex 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 Iadic 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 + I^{n} ⊆ U. This turns R into a topological ring. The Iadic topology is Hausdorff if and only if the intersection of all powers of I is the zero ideal (0).
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