---
tag: NYYV5V2FDD
kind: concept
name: Relatively free algebra
msc: [08B20]
specializes: [GTS4DRRMZR]
sources:
  - url: https://en.wikipedia.org/wiki/Free_object
---

## Summary

A relatively free algebra is an algebra that is free over some of its generating sets in the variety that it generates.
The word "relatively" contrasts it with absolutely free algebras: it is free relative to the identities that it satisfies.
Free groups, free abelian groups, free nilpotent groups of a given class and polynomial rings are relatively free.
Relatively free algebras are exactly the free algebras of all varieties.

## Explanations

### L1 human

An algebra is relatively free when some generators can be sent anywhere inside the algebra itself and the map always extends to an endomorphism.
For example, in $\Z^2$ the generators $(1, 0)$ and $(0, 1)$ can be sent to any two vectors, and a $2 \times 2$ integer matrix does the rest.
So $\Z^2$ is relatively free: it is the free algebra on two generators of the variety of abelian groups.