---
tag: 76BXV761BR
kind: definition
name: Residually finite algebra
msc: [08B26, "20E26"]
defines: X9072A41NK
uses: [KBXSZC6BV1, 88WN3AD2VQ, HEE922EHQ8]
sources:
  - url: https://en.wikipedia.org/wiki/Residual_property_(mathematics)
---

## Definition

Fix a signature and let $A$ be an algebra.
The algebra $A$ is *residually finite* if the following holds for all $a, b \in A$ with $a \ne b$.
There are a finite algebra $F$ of the same signature and a [[homomorphism|KBXSZC6BV1]] $f \colon A \to F$ with $f(a) \ne f(b)$.

## Remarks

Equivalently, $A$ is isomorphic to a [[subdirect product|88WN3AD2VQ]] of finite algebras.
Equivalently, the congruences of $A$ with finite quotient intersect to the trivial congruence.
For groups, this is the definition of a [[residually finite group|HEE922EHQ8]].

Example: the group $(\Z, +)$ is residually finite.
For $a \ne b$, let $n = \abs{a - b} + 1$; the reduction map $\Z \to \Z/n\Z$ sends $a$ and $b$ to different classes, since $n$ does not divide $a - b$.

Non-example: the group $(\Q, +)$ is not residually finite.
A homomorphic image of $\Q$ is divisible, and a finite divisible abelian group is trivial, since multiplication by its order is both zero and onto.
So every homomorphism from $\Q$ to a finite group sends $0$ and $1$ to the same element.