Centralizer of Ring Subset is Subring

Theorem
Let $S$ be a subset of a ring $\struct {R, +, \circ}$

Then $\map {C_R} S$, the centralizer of $S$ in $R$, is a subring of $R$.

If a unit $u \in R$ such that $u \in \map {C_R} S$, then $u^{-1} \in \map {C_R} S$.

Proof
Certainly $0_R \in \map {C_R} S$ as $0_R$ commutes (trivially) with all elements of $R$.

Suppose $x, y \in \map {C_R} S$.

Then:

So:
 * $x + \paren {-y} \in \map {C_R} s$

Suppose $x, y \in \map {C_R} S$ again.

Then from Element Commutes with Product of Commuting Elements:
 * $x \circ y \in \map {C_R} S$

Thus all the conditions are fulfilled for Subring Test, and $\map {C_R} S$ is a subring of $R$.