# 3-locally nilpotent 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: 3-locally nilpotent group

View other analogues of 3-locally nilpotent group | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

## Contents

## Definition

A Lie ring is termed a **3-locally nilpotent Lie ring** if the subring of generated by any subset of of size at most three is a nilpotent Lie ring.

If there is a common bound on the nilpotency class for all such subrings, then the smallest common bound is termed the 3-local nilpotency class.

## Relation with other properties

### Stronger properties

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

abelian Lie ring | LCS-Lazard Lie ring|FULL LIST, MORE INFO | |||

Lie ring of nilpotency class two | |FULL LIST, MORE INFO | |||

nilpotent Lie ring | |FULL LIST, MORE INFO | |||

LCS-Lazard Lie ring | |FULL LIST, MORE INFO | |||

Lazard Lie ring | LCS-Lazard Lie ring|FULL LIST, MORE INFO | |||

locally nilpotent Lie ring | |FULL LIST, MORE INFO |

### Weaker properties

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

Engel Lie ring | |FULL LIST, MORE INFO |