$I$-adic topology

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

$$\bigcap _{k=1}^{\mathrm{\infty }}{I}^k=\{0\}.$$

Though not usually explicitly done, we can define a metric on $R$ by defining $or{d}_I(r)$ for a $r\in R$ by $or{d}_I(r)=k$ 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 $or{d}_I(0)=\mathrm{\infty }$, and then defining for any $r_1,r_2\in R$,

$$d_I(r_1,r_2)=2^{-or{d}_I(r_1-r_s)}.$$

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 $m,n\ge N$ 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 $ϵ$). 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
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