# Verbal 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: verbal subgroup
View other analogues of verbal 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 verbal ideal if it satisfies both the following conditions:

1. It is a verbal Lie subring, i.e., it is the image of a set of words.
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

### Weaker 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
characteristic ideal of a Lie ring ideal that is invariant under all Lie ring automorphisms Fully invariant ideal of a Lie ring|FULL LIST, MORE INFO
ideal of a Lie ring ideal (i.e., invariant under all inner derivations) Characteristic ideal of a Lie ring|FULL LIST, MORE INFO
verbal Lie subring subring (not necessarily ideal) that is generated by words |FULL LIST, MORE INFO
fully invariant Lie subring subring (not necessarily ideal) that is invariant under all Lie ring endomorphisms Fully invariant ideal of a Lie ring|FULL LIST, MORE INFO
characteristic Lie subring subring (not necessarily ideal) that is invariant under all Lie ring automorphisms Characteristic ideal of a Lie ring, Fully invariant ideal of a Lie ring|FULL LIST, MORE INFO