Characteristic of Division Ring is Zero or Prime

Theorem
Let $\left({D, +, \circ}\right)$ be a division ring.

Let $\operatorname{Char} \left({D}\right)$ be the characteristic of $D$.

Then $\operatorname{Char} \left({D}\right)$ is either $0$ or a prime number.

Proof
By definition, a division ring has no proper zero divisors.

If $\left({D, +, \circ}\right)$ is finite, then from Characteristic of Ring with No Zero Divisors, $\operatorname{Char} \left({D}\right)$ is prime.

On the other hand, suppose $\left({D, +, \circ}\right)$ is not finite.

Then there are no $x, y \in D, x \ne 0 \ne y$ such that $x + y = 0$.

Thus it follows that $\operatorname{Char} \left({D}\right)$ is zero.