# 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