Characteristic Lie subring

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
An alternative analogue of characteristic subgroup in Lie ring is: derivation-invariant Lie subring
View other analogues of characteristic subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)

Definition

A subring of a Lie ring is termed a characteristic subring if it is invariant under all automorphisms of the Lie ring.

Relation with properties in related groups

• Lazard correspondence establishes a correspondence between characteristic Lazard Lie subgroups and characteristic Lazard Lie subrings
• Lazard correspondence establishes a correspondence between powering-invariant characteristic subgroups and powering-invariant characteristic subrings

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant Lie subring	Lie subring that is invariant under all endomorphisms			|FULL LIST, MORE INFO
verbal Lie subring	generated by a set of words using the Lie operations			|FULL LIST, MORE INFO

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive Lie subring property	Yes	characteristicity is transitive for Lie rings	Suppose $A\leq B\leq L$ are Lie rings such that $A$ is a characteristic subring of $B$ and $B$ is a characteristic subring of $L$. Then, $A$ is a characteristic subring of $L$.
Lie bracket-closed Lie subring property	Yes	characteristicity is Lie bracket-closed for Lie rings	Suppose $A,B\leq L$ are characteristic subrings. Then, the Lie bracket $[A,B]$ is also a characteristic subring of $L$.