Valuation Ideal is Maximal Ideal of Induced Valuation Ring

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 $\PP$ is an ideal of $\OO$:
 * $(a):\quad \PP$ is a maximal left ideal
 * $(b):\quad \PP$ is a maximal right ideal
 * $(c):\quad$ the quotient ring $\OO / \PP$ is a division ring.

Proof
First it is shown that $\PP$ is an ideal of $\OO$ by applying Test for Ideal.

That is, it is shown that:


 * $(1): \quad \PP \ne \O$
 * $(2): \quad \forall x, y \in \PP: x + \paren {-y} \in \PP$
 * $(3): \quad \forall x \in \PP, y \in \OO: x y \in \PP$

(1)

By :
 * $\norm {0_R} = 0$

Hence:
 * $0_R \in \PP \ne \O$

(2)

Let $x, y \in \PP$.

Then:

Hence:
 * $x + \paren {-y} \in \PP$

(3)

Let $x \in \PP, y \in \OO$.

Then:

Hence:
 * $x y \in \PP$

By Test for Ideal it follows that $\PP$ is an ideal of $\OO$.

By Maximal Left and Right Ideal iff Quotient Ring is Division Ring the statements (a), (b) and (c) above are equivalent.

It is now shown that statement (a) holds.

Let $J$ be a left ideal of $\OO$:
 * $\PP \subsetneq J \subset \OO$

Let $x \in J \setminus \PP$, then:
 * $\norm x = 1$

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, $J = \OO$

Hence $\PP$ is a maximal left ideal.

The result follows.