---
tag: X9072A41NK
kind: concept
name: Residually finite algebra
msc: [08B26, "20E26"]
sources:
  - url: https://en.wikipedia.org/wiki/Residual_property_(mathematics)
---

## Summary

An algebra is residually finite when any two distinct elements can be told apart in some finite homomorphic image.
Equivalently, it is a [[subdirect product|88WN3AD2VQ]] of finite algebras.
For groups this is the notion of a [[residually finite group|HEE922EHQ8]].
Residual finiteness lets one study an infinite algebra through its finite quotients, and it is linked to the solvability of word problems.

## Explanations

### L1 human

Imagine you can only look at finite pictures of an algebra.
The algebra is residually finite if, whenever two elements are different, some finite picture already shows the difference.
The integers are an example: two different integers $a$ and $b$ have different remainders modulo any $n > \abs{a - b}$.
The rationals under addition are not: they have no nontrivial finite quotient, so no finite picture separates $0$ from $1$.