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 $x \in \OO \setminus \PP$.
Aiming for a contradiction, suppose $x \in J$.
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 Definition of Local Ring:
- $\OO$ 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.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.4$ Algebra, Proposition $2.4.1$