# Invariance under any set of derivations is centralizer-closed

## Contents

## Statement

### For a single derivation

Suppose is a Lie ring, is a subring of a Lie ring, and is a derivation of such that . Let be the centralizer of . Then, is also invariant under .

### For a bunch of derivations

Suppose is a Lie ring, is a subring of a Lie ring, and is a set of derivations of such that for all . Let be the centralizer of . Then, is also invariant under all .

## Related facts

### Particular cases

### Analogues in group theory

- Auto-invariance implies centralizer-closed
- Normality is centralizer-closed
- Characteristicity is centralizer-closed

## Proof

Note that the statements for a single derivatoin and for a bunch of derivations are clearly equivalent, so we only prove the former.

**Given**: A Lie ring , a subring of . is the set of such that for all . A derivation of such that .

**To prove**: .

**Proof**: For any and , we need to show that .

For this, note that, by the Leibniz rule property of derivations:

.

Since , the left side is zero. Further, since , , so . This gives as required.