---
tag: N5KE7FH6CX
kind: concept
name: Quotient algebra
msc: [08A30]
sources:
  - url: https://en.wikipedia.org/wiki/Quotient_(universal_algebra)
same_as:
  wikidata: Q2589508
  wikipedia: Quotient (universal algebra)
---

## Summary

A quotient algebra is the algebra of classes of a [[congruence relation|DCSXXT4SER]], with the operations computed on representatives.
It generalizes the [[quotient group|0KYX2N69J0]] and the [[quotient ring|0076HWHGHQ]].
Every homomorphic image of an algebra is isomorphic to one of its quotients, by the [[homomorphism theorem|N783XNJNBV]].

## Explanations

### L1 human

Start with an algebra and a congruence, which says which elements to treat as equal.
Glue each class of equal elements into a single point.
To apply an operation to some classes, pick one element from each class, apply the operation, and take the class of the result.
The congruence guarantees that the choice does not matter.
For the integers with congruence modulo $5$, this gives the five-element ring $\Z/5\Z$.