---
tag: 9CJCXNYNK8
kind: concept
name: Simple algebra
msc: [08A30, 08A05]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_algebra_(universal_algebra)
same_as:
  wikidata: Q1405695
  wikipedia: Simple algebra (universal algebra)
---

## Summary

A simple algebra is a nontrivial [[algebra|QJGPT2W0FT]] whose only [[congruences|DCSXXT4SER]] are the two trivial ones.
So it has no proper nontrivial [[quotient algebras|N5KE7FH6CX]]: every homomorphism out of it is injective or constant.
The notion covers [[simple groups|0YMYMNQ0QZ]] and simple rings, and it is the basic building block in the study of varieties.

## Explanations

### L1 human

A simple algebra cannot be shrunk by gluing elements together, except by gluing everything into one point.
For a group, gluing is controlled by normal subgroups, so a simple group is a group with no normal subgroups other than the trivial one and itself.
For example, $\Z/5\Z$ is a simple group, while $\Z/6\Z$ is not, since it can be glued down to $\Z/2\Z$.
Every field is simple as a ring, because its only ideals are $\set{0}$ and the whole field.