Inverse of Unit in Centralizer of Ring is in Centralizer

Theorem
Let $\struct {R, +, \circ}$ be a ring.

Let $S$ be a subset of $R$.

Let $\map {C_R} S$ denote centralizer of $S$ in $R$

Let $u \in R$ be a unit of $R$.

Then:
 * $u \in \map {C_R} S \implies u^{-1} \in \map {C_R} S$

Proof
Let $u \in R$ be a unit of $R$.

Let $u \in \map {C_R} S$.

Then from Commutation with Inverse in Monoid:
 * $u^{-1} \in \map {C_R} S$