Topological ring

In mathematics, a topological ring is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps: where carries the product topology. That means 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 completion of a topological field may be a topological ring which is not a field. The group of units of a topological ring is a topological group when endowed with the topology coming from the embedding of into the product as However, if the unit group is endowed with the subspace topology as a subspace of it may not be a topological group, because inversion on 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 is continuous in the subspace topology of then these two topologies on 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 (for ) in which multiplication is continuous, too. 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 -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.