---
tag: 2VF2TSJ3PY
kind: concept
name: I-adic topology
msc: [13J10, 13B35]
specializes: [6RF8CB69KQ]
sources:
  - url: https://en.wikipedia.org/wiki/I-adic_topology
same_as:
  wikidata: Q112182519
  wikipedia: I-adic topology
---

## Summary

Let $M$ be a module over a [[commutative ring|0310REJJY8]] $R$, and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.
In the $I$-adic topology on $M$, the submodules $I^n M$ form a base of neighbourhoods of $0$.
Two elements are close when their difference lies in a high power of $I$.
It is the topology behind the [[completion of a ring|KBN13NPYJQ]], and the $p$-adic topology on $\Z$ is the main example.

## Explanations

### L1 human

In this topology an element is small when it is divisible by a high power of $I$.
For the integers and $I = (2)$, the number $2^{10}$ is very small and $1024 + 3$ is very close to $3$.
The powers $2, 4, 8, 16, \dots$ tend to $0$, which never happens for the usual distance on the real line.
The [[definition|MJDV3PFBRT]] makes this precise.