# Characteristic ideal of a Lie ring

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 properties
VIEW 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)

## Definition

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

1. It is a characteristic Lie subring, i.e., it is invariant under all the automorphisms of the whole Lie ring.
2. 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