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 Lie ring

Further information: Lazard Lie ring

Suppose $G$ is a Lazard Lie group and $L$ is its Lazard Lie ring. Under the natural bijection between $L$ and $G$, characteristic subrings of $L$ correspond to characteristic subgroups of $G$.

Metaproperties

Transitivity

This Lie subring property is transitive: a Lie subring with this property in a Lie subring with this property, also has this property.
View a complete list of transitive Lie subring properties

A characteristic Lie subring of a characteristic Lie subring is characteristic in the whole Lie ring. For full proof, refer: Characteristicity is transitive for Lie rings

Lie brackets

This Lie subring property is Lie bracket-closed: the Lie bracket of any two Lie subrings, both with this property, also has this property.
View a complete list of Lie bracket-closed Lie subring properties

The Lie bracket of two characteristic subrings of a Lie ring is again a characteristic subring. For full proof, refer: Characteristicity is Lie bracket-closed for Lie rings

An arbitrary intersection of characteristic Lie subrings is a characteristic Lie subring.

An arbitrary join of characteristic Lie subrings is a characteristic Lie subring.