---
tag: 6DYAGW68AV
kind: concept
name: Semi-local ring
msc: [13H99]
sources:
  - url: https://en.wikipedia.org/wiki/Semi-local_ring
same_as:
  wikidata: Q7449309
  wikipedia: Semi-local ring
---

## Summary

A semi-local ring is a [[commutative ring|0310REJJY8]] with only finitely many [[maximal ideals|0PV74TR5FP]].
It generalizes a [[local ring|MZ02BKPHJA]], which has exactly one.
Semi-local rings arise when one localizes at the complement of a finite union of prime ideals.
They also appear in the study of integral extensions of local rings.

## Explanations

### L1 human

A local ring looks at one point; a semi-local ring looks at finitely many points at once.
For example, the fractions $a/b$ with $b$ prime to both $2$ and $3$ form a ring in which only the primes $2$ and $3$ remain non-invertible.
This ring has exactly two maximal ideals, generated by $2$ and by $3$.
The [[definition|Q256RTTNTN]] gives the condition.