Invariance under any set of derivations is centralizer-closed
For a single derivation
For a bunch of derivations
Analogues in group theory
- Auto-invariance implies centralizer-closed
- Normality is centralizer-closed
- Characteristicity is centralizer-closed
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.