Characteristic Lie subring

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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: 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 Lie subring or characteristic subring if it is invariant under all automorphisms of the Lie ring.

Relation with properties in related groups

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
fully invariant subgroup of additive group of a Lie ring subset of a Lie ring that is invariant under all endomorphisms of the additive group of the Lie ring Fully invariant Lie subring, Fully invariant ideal of a Lie ring, Lie subring invariant under any additive endomorphism satisfying a comultiplication condition|FULL LIST, MORE INFO
verbal Lie subring generated by a set of words using the Lie operations |FULL LIST, MORE INFO
marginal Lie subring |FULL LIST, MORE INFO
strictly characteristic Lie subring |FULL LIST, MORE INFO
injective endomorphism-invariant Lie subring |FULL LIST, MORE INFO}
characteristic ideal of a Lie ring characteristic and also an ideal |FULL LIST, MORE INFO
characteristic derivation-invariant Lie subring characteristic and also a derivation-invariant Lie subring |FULL LIST, MORE INFO

Incomparable properties

Property Meaning Proof of forward non-implication Proof of reverse non-implication Conjunction
ideal of a Lie ring invariant under all inner derivations characteristic not implies ideal ideal not implies characteristic characteristic ideal of a Lie ring
derivation-invariant Lie subring invariant under all derivations characteristic not implies derivation-invariant derivation-invariant not implies characteristic characteristic derivation-invariant Lie subring

Metaproperties

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