# Fully invariant 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: fully invariant subgroup

View other analogues of fully invariant 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 **fully invariant Lie subring** if, for every endomorphism of , .

### Lazard Lie ring

Suppose is a Lazard Lie group and is a Lazard Lie ring. Under the natural bijection between and , fully invariant subrings of correspond to fully invariant subgroups of .

## Relation with other properties

### Stronger properties

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

Fully invariant subgroup of additive group of a Lie ring | Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant | |||

Lie subring invariant under any additive endomorphism satisfying a comultiplication condition |

### Weaker properties

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

Characteristic subring of a Lie ring |

## Metaproperties

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

transitive Lie subring property | Yes | full invariance is transitive for Lie rings | If are Lie rings, with fully invariant in and fully invariant in , then is fully invariant in . |

Lie bracket-closed Lie subring property | Yes | full invariance is Lie bracket-closed for Lie rings | If are fully invariant Lie subrings of a Lie ring , so is . |

strongly intersection-closed Lie subring property | Yes | full invariance is strongly intersection-closed for Lie rings | If are all fully invariant Lie subrings of a Lie ring , then so is . |