If $R$ is a ring, let $Z(R) = \{x ∈ R \mid xy = yx \forall y ∈ R\}$. Prove that $Z(R)$ is a subring of $R$.
Clearly $0 ∈ Z(R)$, so $Z(R) \neq ∅$. Let $x, y ∈ Z(R)$ and let $r ∈ R$. $(x − y)r = xr − yr$. Since $x$ and $y$ are in the center this is equal to $rx − ry = r(x − y)$. So $x − y ∈ Z(R)$. Now $(xy)r = x(yr) = x(ry) =(xr)y = r(xy)$ for all $r ∈ R.$
Hence $xy ∈ Z(R)$.
Is there any other way to demonstrate this?
Thank you in advance.
