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

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 $L$.