Derivation-invariant Lie subring

This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring
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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
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Definition

A subring $A$ of a Lie ring $L$ is termed derivation-invariant if $d(A)\subseteq A$ for every derivation $d$ of $L$.

Relation with other properties

Stronger properties

In some special circumstances, any characteristic subring of a Lie ring is derivation-invariant. This happens when every derivation can be exponentiated to an automorphism of the Lie ring.

Weaker properties

• Ideal: This is a subring invariant under all inner derivations. The fact that any derivation-invariant subring is an ideal, is analogous to the fact that characteristic implies normal