Center is local powering-invariant in Lie ring

Statement

Suppose $L$ is a Lie ring and $Z(L)$ is its center. Then, $Z(L)$ is a local powering-invariant subring of $L$. Explicitly, if $a\in Z(L)$ and $n$ is a natural number such the equation $nx=a$ has a unique solution for $x\in L$, then that unique solution $x$ is in $Z(L)$.

Related facts

• Center is local powering-invariant
• Derived subring is divisibility-invariant in Lie ring