# Characteristic Lie subring

## 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 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
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 $L$.