---
tag: TEV7XV14F5
kind: definition
name: Finite ring
msc: [16P10, 13M05]
defines: X6713ZP2NH
uses: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_ring
---

## Definition

Let $R$ be a [[ring|0B1261B5ED]].
The ring $R$ is a *finite ring* if its underlying set is finite.
The number of elements of $R$ is the *order* of $R$, written $\abs{R}$.

## Remarks

For each integer $n \ge 1$, the ring $\Z/n\Z$ of [[integers modulo n|03VFZ2WETT]] is a finite ring of order $n$, since there are exactly $n$ residue classes.
The ring $M_2(\F_2)$ of $2 \times 2$ matrices over the [[finite field|0ZXZ82FTX2]] $\F_2$ is a finite ring of order $2^4 = 16$, and it is not commutative.
The ring $\Z$ of integers is not a finite ring, because it has infinitely many elements.
Every finite [[integral domain|0QVD7WBFGN]] is a field: multiplication by a nonzero element is injective, hence bijective on a finite set, so every nonzero element has an inverse.