---
tag: X6713ZP2NH
kind: concept
name: Finite ring
msc: [16P10, 13M05]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_ring
same_as:
  wikidata: Q2354159
  wikipedia: Finite ring
---

## Summary

A finite ring is a [[ring|0B1261B5ED]] with only finitely many elements.
The rings $\Z/n\Z$ of [[integers modulo n|03VFZ2WETT]], the [[finite fields|0ZXZ82FTX2]] and the matrix rings over them are the basic examples.
Finiteness forces strong structure: by [[Wedderburn's little theorem|0RRNVX1VKA]], every finite division ring is a field.
Finite rings are used in number theory, combinatorics and coding theory.

## Explanations

### L1 human

A finite ring is a number system with only finitely many numbers, where you can add, subtract and multiply.
Clock arithmetic is the main picture: on a clock with $12$ hours, $9 + 5 = 2$ and $3 \cdot 4 = 0$.
This gives the ring $\Z/12\Z$ with $12$ elements.
In a finite ring, repeated addition of $1$ must come back to $0$, so the ring always has a positive characteristic.