# Characteristic Lie subring

From Groupprops

This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring

View a complete list of such propertiesVIEW RELATED: Lie subring property implications | Lie subring property non-implications | Lie subring metaproperty satisfactions | Lie subring metaproperty dissatisfactions | Lie subring property satisfactions |Lie subring property dissatisfactions

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

An alternative analogue of characteristic subgroup in Lie ring is: derivation-invariant Lie subring

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

## Contents

## Definition

A subring of a Lie ring is termed a **characteristic Lie subring** or **characteristic subring** if it is invariant under all automorphisms of the Lie ring.

- Lazard correspondence establishes a correspondence between characteristic Lazard Lie subgroups and characteristic Lazard Lie subrings
- Lazard correspondence establishes a correspondence between powering-invariant characteristic subgroups and powering-invariant characteristic subrings

## Relation with other properties

### Stronger properties

### Incomparable properties

Property | Meaning | Proof of forward non-implication | Proof of reverse non-implication | Conjunction |
---|---|---|---|---|

ideal of a Lie ring | invariant under all inner derivations | characteristic not implies ideal | ideal not implies characteristic | characteristic ideal of a Lie ring |

derivation-invariant Lie subring | invariant under all derivations | characteristic not implies derivation-invariant | derivation-invariant not implies characteristic | characteristic derivation-invariant Lie subring |

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive Lie subring property | Yes | characteristicity is transitive for Lie rings | Suppose are Lie rings such that is a characteristic subring of and is a characteristic subring of . Then, is a characteristic subring of . |

Lie bracket-closed Lie subring property | Yes | characteristicity is Lie bracket-closed for Lie rings | Suppose are characteristic subrings. Then, the Lie bracket is also a characteristic subring of . |