# Ideal property is centralizer-closed

From Groupprops

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

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## Contents

## Statement

Suppose is a Lie ring, is an ideal of , and is the centralizer of in . Then, is also an ideal of .

## Related facts

### Analogue for group theory

### Generalizations

The general version of this result is: invariance under any set of derivations is centralizer-closed, which reduces to this result when we take the inner derivations (see Lie ring acts as derivations by adjoint action). Another special case of this general result is:

## Proof

### Proof using the notion of inner derivations

### Hands-on proof using Jacobi identity

**Given**: A Lie ring , an ideal of . is the centralizer of in .

**To prove**: is an ideal of , i.e., for any and , .

**Proof**: To show it suffices to show that for all .

By the Jacobi identity:

.

Since centralizes , , so the second term is zero. Further, since is an ideal, , and since centralizes , . Thus, both the second and third term on the left side are zero, forcing .