---
tag: G9Q61B8A7C
kind: concept
name: Complete local ring
msc: [13J10]
specializes: [MZ02BKPHJA]
sources:
  - url: https://en.wikipedia.org/wiki/Completion_of_a_ring
---

## Summary

A complete local ring is a Noetherian [[local ring|MZ02BKPHJA]] $(R, \mathfrak m)$ that equals its own [[completion|KBN13NPYJQ]] for the $\mathfrak m$-adic topology.
Formal power series rings over a field and the $p$-adic integers are the basic examples.
Complete local rings have a clear structure, given by the [[Cohen structure theorem|N3XH42KVV3]], and every Noetherian local ring embeds into a complete one, its completion.

## Explanations

### L1 human

In a complete local ring, every list of better and better approximations has a limit.
For example, in the formal power series ring $k[\![x]\!]$ the polynomials $1$, $1 + x$, $1 + x + x^2$, $\dots$ approach the series $1 + x + x^2 + \cdots$, which is again an element of the ring.
In the ring of rational functions defined at $0$, many such lists have no limit, so that ring is not complete.
The [[definition|78GT1VFFPR]] makes this precise.