# $I$-adic topology

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

 $\bigcap_{k=1}^{\infty}I^{k}=\{0\}.$

Though not usually explicitly done, we can define a metric on $R$ by defining $ord_{I}(r)$ for a $r\in R$ by $ord_{I}(r)=k$ where $k$ is the largest integer such that $r\in I^{k}$ (well-defined by the intersection assumption, and $I^{0}$ is taken to be the entire ring) and by $ord_{I}(0)=\infty$, and then defining for any $r_{1},r_{2}\in R$,

 $d_{I}(r_{1},r_{2})=2^{-ord_{I}(r_{1}-r_{s})}.$

The topology 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 equivalent 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 $\{r_{i}\}\in R$ is Cauchy with respect to this topology if for any $k$ there exists an $N$ such that for all $m,n\geq N$ we have $(a_{m}-a_{n})\in I^{k}$. (Note the parallel with the metric version of Cauchy, where $k$ plays the part analogous to an arbitrary $\epsilon$). The ring $R$ is complete with respect to the $I$-adic topology if every such Cauchy sequence converges to an element of $R$.

Title $I$-adic topology IadicTopology 2013-03-22 14:36:59 2013-03-22 14:36:59 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 13B35 I-adic topology