Verbal ideal of a Lie ring

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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