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.