Prove that the center of a division ring is a field.

Proof. Recall that the center of a ring R is defined as:

\begin{align*}
Z = \{z \in R \ | \ zr = rz \ \text{for all} \ r \in R\}.
\end{align*}

Now we have that R is a division ring, and 1 is contained in Z. It is also easy to see that Z is a division ring, since Z is a subring and the elements that are contained in Z are units, except for the 0. Since Z is commutative, we see that Z is a commutative division ring, which is called a field.
So, the center of a division ring is a field.