I-adic topology

Let R be a ring and I an ideal in R such that


Though not usually explicitly done, we can define a metric on R by defining ordI(r) for a rR by ordI(r)=k where k is the largest integer such that rIk (well-defined by the intersectionDlmfMathworldPlanetmath assumptionPlanetmathPlanetmath, and I0 is taken to be the entire ring) and by ordI(0)=, and then defining for any r1,r2R,


The topologyMathworldPlanetmath induced by this metric is called the I-adic topology. Note that the number 2 was chosen rather arbitrarily. Any other real number greater than 1 will induce an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath topology.

Except in the case of the similarly-defined p-adic topology, it is rare that reference is made to the actual I-adic metric. Instead, we usually refer to the I-adic topology.

In particular, a sequence of elements in {ri}R is Cauchy with respect to this topology if for any k there exists an N such that for all m,nN we have (am-an)Ik. (Note the parallel with the metric version of Cauchy, where k plays the part analogous to an arbitrary ϵ). The ring R is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with respect to the I-adic topology if every such Cauchy sequencePlanetmathPlanetmath convergesPlanetmathPlanetmath to an element of R.

Title I-adic topology
Canonical name IadicTopology
Date of creation 2013-03-22 14:36:59
Last modified on 2013-03-22 14:36:59
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 13B35
Synonym I-adic topology