# Derivation-invariance is centralizer-closed

From Groupprops

This article gives the statement, and possibly proof, of a Lie subring property (i.e., derivation-invariant subring) satisfying a Lie subring metaproperty (i.e., centralizer-closed Lie subring property)

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ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: characteristicity is centralizer-closed.

Another analogue to the same fact, in the same new context, is: characteristicity is centralizer-closed for Lie rings

View other analogues of characteristicity is centralizer-closed|View other analogues from group to Lie ring (OR, View as a tabulated list)

## Statement

### Verbal statement

The centralizer of a derivation-invariant subring of a Lie ring is also derivation-invariant.

## Related facts

### Weaker facts

## Proof

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

**To prove**: is also derivation-invariant, i.e., for any .

**Proof**: Suppose . 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 is derivation-invariant, , so . This gives as required.