Let be a ring and an ideal in such that
Though not usually explicitly done, we can define a metric on by defining for a by where is the largest integer such that (well-defined by the intersection assumption, and is taken to be the entire ring) and by , and then defining for any ,
Except in the case of the similarly-defined -adic topology, it is rare that reference is made to the actual -adic metric. Instead, we usually refer to the -adic topology.
In particular, a sequence of elements in is Cauchy with respect to this topology if for any there exists an such that for all we have . (Note the parallel with the metric version of Cauchy, where plays the part analogous to an arbitrary ). The ring is complete with respect to the -adic topology if every such Cauchy sequence converges to an element of .
|Date of creation||2013-03-22 14:36:59|
|Last modified on||2013-03-22 14:36:59|
|Last modified by||mathcam (2727)|