# Zelmanov's theorem on Engel Lie rings

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term locally nilpotent Lie ring

View a complete list of pages giving proofs of equivalence of definitions

## Statement

The statement has many equivalent versions:

- For any natural number , any finitely generated -Engel Lie ring is a nilpotent Lie ring. Note that finite generation is
*as a Lie ring*, not necessarily as an abelian group. - For any natural number , any -Engel Lie ring is a locally nilpotent Lie ring.
- A Lie ring is a locally nilpotent Lie ring if and only it is a (not necessarily bounded) Engel Lie ring.

## Related facts

### Similar facts for Lie rings

- Engel's theorem is a similar result for finite-dimensional Lie algebras over fields
- 2-Engel implies class three for Lie rings
- 2-Engel and 3-torsion-free implies class two for Lie rings
- 3-Engel and (2,5)-torsion-free implies class six for Lie rings

### Similar facts for groups

- Zorn's theorem says that finite Engel groups are nilpotent
- 2-Engel implies class three for groups
- 2-Engel and 3-torsion-free implies class two for groups
- 3-Engel and (2,5)-torsion-free implies class four for groups
- 3-Engel implies locally nilpotent for groups
- 4-Engel and (2,3,5)-torsion-free implies class seven for groups
- 4-Engel implies locally nilpotent for groups
- The questions are open for -Engel groups, .