---
tag: EHASX3TBRQ
kind: concept
name: Abelian algebra
msc: [08A30, 08B10]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_algebra
---

## Summary

An abelian algebra is an algebra in which the [[commutator|6DSWPHSWQW]] of the full congruence with itself is the trivial congruence.
Equivalently, its term operations satisfy the term condition with equality.
For groups this notion gives exactly the abelian groups, and modules over a ring are abelian algebras.
Abelian algebras play the role of abelian groups in the general theory of solvable and nilpotent algebras.

## Explanations

### L1 human

In an abelian group, changing one input of a formula has an effect that does not depend on the other inputs.
An abelian algebra is an algebra with the same property for every formula built from its operations.
Modules and vector spaces are abelian in this sense.
Nonabelian groups and lattices with more than one element are not.