# Characteristic ideal of a Lie ring

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

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

## Contents

## Definition

A subset of a Lie ring is termed a **characteristic ideal** if it satisfies **both** the following conditions:

- It is a characteristic Lie subring, i.e., it is invariant under all the automorphisms of the whole Lie ring.
- It is an ideal of the Lie ring, i.e., it is invariant under all the inner derivations of the whole Lie ring.

## Relation with other properties

### Stronger properties

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

fully invariant ideal of a Lie ring | ideal that is invariant under all Lie ring endomorphisms | |FULL LIST, MORE INFO | ||

verbal ideal of a Lie ring | ideal that is generated by a set of words. | Fully invariant ideal of a Lie ring|FULL LIST, MORE INFO |

### Weaker properties

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

characteristic Lie subring | invariant under all Lie ring automorphisms | characteristic subring not implies ideal | |FULL LIST, MORE INFO | |

ideal of a Lie ring | invariant under all inner derivations | |FULL LIST, MORE INFO |