---
tag: NS84D2920R
kind: definition
name: Completion of a ring
msc: [13B35, 13J10]
defines: KBN13NPYJQ
uses: [0310REJJY8, 096Z9MP2ME, 0076HWHGHQ, 2VF2TSJ3PY]
sources:
  - url: https://en.wikipedia.org/wiki/Completion_of_a_ring
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.
For $m \ge n$ let $\pi_{mn} \colon R/I^m \to R/I^n$ be the natural map of [[quotient rings|0076HWHGHQ]].
The *$I$-adic completion* of $R$ is the inverse limit
$$\hat R = \varprojlim_n R/I^n = \set{(x_n)_{n \ge 1} \in \prod_{n \ge 1} R/I^n : \pi_{mn}(x_m) = x_n \text{ for all } m \ge n},$$
with componentwise addition and multiplication.

## Remarks

The map $R \to \hat R$, $r \mapsto (r + I^n)_n$, is a ring homomorphism with kernel $\bigcap_n I^n$.
Equivalently, $\hat R$ is the ring of Cauchy sequences for the [[$I$-adic topology|2VF2TSJ3PY]] modulo the sequences that tend to $0$.
For an $R$-module $M$ one defines $\hat M = \varprojlim_n M/I^n M$ in the same way.
Example: for $R = k[x]$, with $k$ a field, and $I = (x)$, a compatible sequence is the list of truncations of a formal power series, and $\hat R \cong k[\![x]\!]$.
Non-example: for $R = \Z$ and $I = 2\Z$, the sequence starting with $x_1 = 0 + 2\Z$ and $x_2 = 1 + 4\Z$ does not give an element of $\hat R$, since $\pi_{21}(x_2) = 1 + 2\Z \neq x_1$.