# Lie subring invariant under any derivation with partial divided Leibniz condition powers

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

## Definition

Suppose $L$ is a Lie ring and $S$ is a Lie subring of $L$. We say that $S$ is invariant under any derivation with partial divided Leibniz condition powers if the following holds: for any positive integer $m$ and any derivation with divided Leibniz condition powers up to $m$ for $L$ given by $d^{(1)}, d^{(2)}, \dots, d^{(m)}$, we have $d^{(i)}(S) \subseteq S$ for all $i \in \{ 1,2,\dots,m\}$.

## Relation with other properties

### Weaker properties

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