Lie subring invariant under any additive endomorphism satisfying a comultiplication condition

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

Statement

Suppose $L$ is a Lie ring and $S$ is a Lie subring of $L$. We say that $S$ is invariant under any additive endomorphism satisfying a comultiplication condition if, for any $f:L\to L$ that is an additive endomorphism satisfying a comultiplication condition, we have $f(S)\subseteq S$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant subgroup of additive group of a Lie ring

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant Lie subring
derivation-invariant Lie subring
characteristic subring of a Lie ring