Editing Topological ring (section)

==Completion==
{{main article|Completion (algebra)}}

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 ring <math>R</math> is [[Complete uniform space|complete]]. If it is not, then it can be ''completed'': one can find an essentially unique complete topological ring <math>S</math> that contains <math>R</math> as a [[Dense (topology)|dense]] [[subring]] such that the given topology on <math>R</math> equals the [[Subspace (topology)|subspace topology]] arising from <math>S.</math>
If the starting ring <math>R</math> is metric, the ring <math>S</math> can be constructed as a set of equivalence classes of [[Cauchy sequence]]s in <math>R,</math> this equivalence relation makes the ring <math>S</math> Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) <math>c : R \to S</math> such that, for all CM <math>f : R \to T</math> where <math>T</math> is Hausdorff and complete, there exists a unique CM <math>g : S \to T</math> such that 
<math>f = g \circ c.</math> If <math>R</math> is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions <math>f : \R \to \Q</math> endowed with the topology of pointwise convergence) the standard construction uses minimal Cauchy filters and satisfies the same universal property as above (see [[Nicolas Bourbaki|Bourbaki]], General Topology, III.6.5).

The rings of [[formal power series]] and the [[p-adic number|<math>p</math>-adic integers]] are most naturally defined as completions of certain topological rings carrying [[I-adic topology|<math>I</math>-adic topologies]].