# Engel Lie ring

From Groupprops

This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.

View a complete list of properties of Lie ringsVIEW RELATED: Lie ring property implications | Lie ring property non-implications |Lie ring metaproperty satisfactions | Lie ring metaproperty dissatisfactions | Lie ring property satisfactions | Lie ring property dissatisfactions

ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie ring property analogous to the group property: Engel group

View other analogues of Engel group | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

## Contents

## Definition

### Definition in terms of inner derivations

A Lie ring is termed a -**Engel Lie ring** if every inner derivation of the Lie ring is a nilpotent derivation with nilpotency at most .

### Definition in terms of Lie brackets

A Lie ring is termed a -**Engel Lie ring** if there exists a natural number such that for any , t:

where occurs times.

Note that we sometimes simply use the term **Engel Lie ring** for a Lie ring where the value of may be dependent on and . However, by Zelmanov's theorem on Engel Lie rings, this is equivalent to being a locally nilpotent Lie ring.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

nilpotent Lie ring | ||||

Lazard Lie ring | ||||

inner-Lazard Lie ring | ||||

Lie ring in which every inner derivation is exponentiable | every inner derivation is an exponentiable derivation |

## Facts

- Engel's theorem: This states that if is a finite-dimensional Lie algebra over a field, the Engel property is equivalent to being nilpotent.
- Zelmanov's theorem on Engel Lie rings: This states that Engel Lie rings are locally nilpotent.
- Kostrikin's theorem: This states that if is a -generator Lie algebra over a field of characteristic satisfying the -Engel condition for (or arbitrary in the case ) then is nilpotent with nilpotency class bounded by a function of and .