Centralizer of Ring Subset is Subring

Theorem
Let $S$ be a subset of a ring $\left({R, +, \circ}\right)$

Then $C_R \left({S}\right)$, the centralizer of $S$ in $R$, is a subring of $R$.

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

Proof

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


 * Suppose $x, y \in C_R \left({S}\right)$. Then:

So $x + \left({-y}\right) \in C_R \left({S}\right)$.


 * Suppose $x, y \in C_R \left({S}\right)$ again. Then: $x \circ y \in C_R \left({S}\right)$ from Associativity and Commutativity Properties.

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


 * Let $u \in U_R$.

If $u \in C_R \left({S}\right)$, then $u^{-1} \in C_R \left({S}\right)$ from Commutation with Inverse.