Topological ring
In mathematics, a topological ring is a ring formula_1 that is also a topological space such that both the addition and the multiplication are continuous as maps: formula_2 where formula_3 carries the product topology. That means formula_1 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. General comments. The group of units formula_5 of a topological ring formula_1 is a topological group when endowed with  the topology coming from the embedding of formula_5 into the product formula_3 as formula_9 However, if the unit group is endowed with the subspace topology as a subspace of formula_10 it may not be a topological group, because inversion on formula_5 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 formula_5 is continuous in the subspace topology of formula_1 then these two topologies on formula_5 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 formula_15) in which multiplication is continuous, too. Examples. 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 formula_16 -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 formula_1 containing an ideal formula_18 and then considers the formula_19-adic topology on formula_1: a subset formula_21 of formula_1 is open if and only if for every formula_23 there exists a natural number formula_24 such that formula_25 This turns formula_1 into a topological ring. The formula_19-adic topology is Hausdorff if  and only if the intersection of all powers of formula_19 is the zero ideal formula_29 The formula_16-adic topology on the integers is an example of an formula_19-adic topology (with formula_32). Completion. Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ringformula_1 is complete. If it is not, then it can be "completed": one can find an essentially unique complete topological ring formula_34 that contains formula_1 as a dense subring such that the given topology on formula_1 equals the subspace  topology arising from formula_37 If the starting ring formula_1 is metric, the ring formula_34 can be constructed as a set of equivalence classes of Cauchy sequences in formula_10 this equivalence relation makes the ring formula_34 Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) formula_42 such that, for all CM formula_43 where formula_44 is Hausdorff and complete, there exists a unique CM formula_45 such that formula_46 If formula_1 is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions formula_48 endowed with the  topology of pointwise convergence) the standard construction uses minimal Cauchy filters and satisfies the same universal property as above (see Bourbaki, General Topology, III.6.5). The rings of formal power series and the formula_16-adic integers are most naturally defined as completions of certain topological rings carrying formula_19-adic topologies. Topological fields. Some of the most important examples are topological fields. A topological field is a topological ring that is also a field, and such that inversion of non zero elements is a continuous function. The most common examples are the complex numbers and all its subfields, and the valued fields , which include the formula_16-adic fields.