# Ideal property is centralizer-closed

## Statement

Suppose $L$ is a Lie ring, $S$ is an ideal of $S$, and $C = C_L(S)$ is the centralizer of $S$ in $L$. Then, $C$ is also an ideal of $S$.

## Related facts

### 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

### Hands-on proof using Jacobi identity

Given: A Lie ring $L$, an ideal $S$ of $L$. $C = C_L(S)$ is the centralizer of $S$ in $L$.

To prove: $C$ is an ideal of $L$, i.e., for any $x \in L$ and $c \in C$, $[x,c] \in C$.

Proof: To show $[x,c] \in C$ it suffices to show that $[[x,c],s] = 0$ for all $s \in S$.

By the Jacobi identity:

$[[x,c],s] + [[c,s],x] + [[s,x],c] = 0$.

Since $C$ centralizes $S$, $[c,s] = 0$, so the second term is zero. Further, since $S$ is an ideal, $[s,x] \in S$, and since $C$ centralizes $S$, $[[s,x],c] = 0$. Thus, both the second and third term on the left side are zero, forcing $[[x,c],s] = 0$.