---
tag: MZ02BKPHJA
kind: concept
name: Local ring
msc: [13H99, 13B30]
specializes: [0310REJJY8]
sources:
  - url: https://en.wikipedia.org/wiki/Local_ring
same_as:
  wikidata: Q1142704
  wikipedia: Local ring
---

## Summary

A local ring is a [[commutative ring|0310REJJY8]] with exactly one [[maximal ideal|0PV74TR5FP]].
Equivalently, its non-[[units|0XVEZDE9J8]] form an ideal.
Local rings arise as [[localizations at prime ideals|X6JD4AG973]] and describe the behaviour of a ring, or of a geometric space, near one point.
Many theorems of commutative algebra are first proved for local rings, where [[Nakayama's lemma|15JHBYR6KH]] is a strong tool.

## Explanations

### L1 human

A local ring is a ring that "sees only one point".
A typical example is the ring of rational numbers whose denominators are odd: an element is invertible exactly when its numerator is odd too.
The elements that are not invertible, those with even numerator, form one ideal, the maximal ideal.
Another example is the ring of formal power series $k[\![x]\!]$, where a series is invertible exactly when its constant term is nonzero.
The [[definition|WJS9TPSGVM]] gives the condition and its equivalent forms.