Valuation Ideal is Maximal Ideal of Induced Valuation Ring/Corollary 1

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$.

Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is:
 * $\OO = \set {x \in R : \norm x \le 1}$

Let $\PP$ be the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is:
 * $\PP = \set {x \in R : \norm x < 1}$

Then:
 * $(a): \quad \OO$ is a local ring.
 * $(b): \quad \PP$ is the unique maximal left ideal of $\OO$
 * $(c): \quad \PP$ is the unique maximal right ideal of $\OO$

Proof
By Valuation Ideal is Maximal Ideal of Induced Valuation Ring then:
 * $\PP$ is a maximal left ideal of $\OO$.

Let $J \subsetneq \OO$ be any maximal left ideal of $\OO$.

Let $x \in \OO \setminus \PP$.

$x \in J$.

By Norm of Inverse then:
 * $\norm {x^{-1}} = 1 / \norm x = 1 / 1 = 1$

Hence:
 * $x^{-1} \in \OO$

Since $J$ is a left ideal then:
 * $x^{-1} x = 1_R \in J$

Thus:
 * $\forall y \in \OO: y \cdot 1_R = y \in J$

That is:
 * $J = \OO$

This contradicts the assumption that $J \ne \OO$.

So:
 * $x \notin J$

Hence:
 * $\paren {\OO \setminus \PP} \cap J = \O$

That is:
 * $J \subseteq \PP$

Since $J$ and $\PP$ are both maximal left ideals then:
 * $J = \PP$

Hence:
 * $\PP$ is the unique maximal left ideal of $\OO$.

By :
 * $\OO$ is a local ring

From Maximal Left and Right Ideal iff Quotient Ring is Division Ring:
 * every maximal right ideal of $\OO$ is a maximal left ideal

Hence:
 * $\PP$ is the unique maximal right ideal of $\OO$.

The result follows.