---
tag: 2P3Y2FRCSZ
kind: concept
name: Free algebra
msc: [16S10, 16S15]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Free_algebra
same_as:
  wikidata: Q5500180
  wikipedia: Free algebra
---

## Summary

The free algebra $k\langle X \rangle$ on a set $X$ over a [[commutative ring|0310REJJY8]] $k$ is the algebra of polynomials in noncommuting variables from $X$.
It is the free object among [[associative algebras|3CTW1P4HBP]] over $k$: any map from $X$ into an algebra extends uniquely to an algebra homomorphism.
So every algebra generated by a set $X$ is a quotient of $k\langle X \rangle$, and algebras are often given by generators and relations in this way.
It is the noncommutative analogue of the [[polynomial ring|0SDC934R52]] and is isomorphic to the [[tensor algebra|P4J5JZDW0B]] of the free module with basis $X$.

## Explanations

### L1 human

Take letters $x, y$ and form all words in them, such as $xy$, $yx$, $xxy$.
Unlike in ordinary polynomials, $xy$ and $yx$ are different words.
The free algebra consists of finite linear combinations of words, like $3xy - 2yx + 5$, multiplied by joining words end to end.
Every other algebra with two chosen elements is obtained from this one by imposing rules, for example $xy = yx$ gives back the polynomial ring $k[x, y]$.