Valuation Ring of Non-Archimedean Division Ring is Subring

Theorem
Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$.

Let $0_R$ be the zero of $R$ and $1_R$ be the unity of $R$.

Let
 * $\mathcal O = {B_1}^- \paren {0_R} = \set{x \in R : \norm{x} \le 1}$

where ${B_1}^- \paren {0_R}$ denotes the closed ball with center $0_R$ and radius $1$

Then $\mathcal O$ is a subring of $R$:
 * with a unity $1_R$
 * in which there are no (proper) zero divisors, that is:
 * $\forall x, y \in \mathcal O: x \circ y = 0_R \implies x = 0_R \text{ or } y = 0_R$