---
tag: PZ41AE18X4
kind: concept
name: Projective algebra
msc: [08B30, 18G05]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_object
---

## Summary

An algebra in a class of algebras is projective when every homomorphism from it into a quotient lifts through the quotient map.
It is the universal-algebra form of a projective object, with surjective homomorphisms in the role of epimorphisms.
In a variety, the projective algebras are exactly the retracts of free algebras.
For modules they are the [[projective modules|BFFA03GE3V]].

## Explanations

### L1 human

A projective algebra can always be lifted back along an onto map.
If its image lands in a quotient, one can choose preimages for everything in a way that respects the operations.
Free algebras are projective, because one only needs to choose a preimage for each free generator.
The group $\Z/2\Z$ is not projective among abelian groups: there is no way to lift the identity of $\Z/2\Z$ through $\Z \to \Z/2\Z$, since $\Z$ has no element of order $2$.