Center is derivation-invariant

Statement

Verbal statement

The center of a Lie ring is a derivation-invariant subring.

Statement with symbols

Let $L$ be a Lie ring and $Z$ be its center:

$Z=\{x\in L\mid [x,y]=0\ \forall \ y\in L\}$.

Then, if $d:L\to L$ is a derivation, $d(Z)\subseteq Z$.

Related facts

• Center is invariant under any derivation with partial divided Leibniz condition powers

Proof

Given: A Lie ring $L$, its center $Z=\{x\in L\mid [x,y]=0\ \forall \ y\in L\}$, a derivation $d:L\to L$.

To prove: If $x\in Z$, $dx\in Z$.

Proof: Pick any $y\in L$. Then, we have:

$d[x,y]=[dx,y]+[x,dy]$.

Now, since $x\in Z$, $[x,y]=[x,dy]=0$. Thus, $d[x,y]=[x,dy]=0$, yielding $[dx,y]=0$.

Thus, $[dx,y]=0\ \forall \ y\in L$, so $dx\in Z$.