# Verbal 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: 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:

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