# LUCS-Baer Lie ring

From Groupprops

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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

## Definition

A Lie ring is termed a **LUCS-Baer Lie ring** if it is a Lie ring of nilpotency class two (i.e., the derived subring is contained in the center) and:

- Every element of its derived subring has a unique half in the whole Lie ring.
- Every element of its derived subring has a unique half among the elements in the center of the whole Lie ring.
- Every element of its derived subring has a unique half among the elements in the Lie ring
*and*that unique half is in the center.

## Relation with other properties

### Stronger properties

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

Baer Lie ring | 2-powered Lie ring of nilpotency class two | |FULL LIST, MORE INFO |

### Weaker properties

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

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

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