Valuation Ring of Non-Archimedean Division Ring is Subring
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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:
- $\OO = \set {x \in R : \norm{x} \le 1}$
Then $\OO$ is a subring of $R$:
- with a unity: $1_R$
- in which there are no (proper) zero divisors, that is:
- $\forall x, y \in \OO: x \circ y = 0_R \implies x = 0_R \text{ or } y = 0_R$
Proof
To show that $\OO$ is a subring the Subring Test is used by showing:
- $(1): \quad \OO \ne \O$
- $(2): \quad \forall x, y \in \OO: x + \paren {-y} \in \OO$
- $(3): \quad \forall x, y \in \OO: x y \in \OO$
(1)
By Norm of Unity,
- $\norm{1_R} = 1$
Hence:
- $1_R \in \OO \ne \O$
$\Box$
(2)
Let $x, y \in \OO$.
Then:
\(\ds \norm {x + \paren{-y} }\) | \(\le\) | \(\ds \max \set {\norm x, \norm{-y} }\) | Non-Archimedean Norm Axiom $\text N 4$: Ultrametric Inequality | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\norm x, \norm y}\) | Norm of Negative | |||||||||||
\(\ds \) | \(\le\) | \(\ds 1\) | Since $x, y \in \OO$ |
Hence:
- $x + \paren {-y} \in \OO$
$\Box$
(3)
Let $x, y \in \OO$.
Then:
\(\ds \norm{x y}\) | \(\le\) | \(\ds \norm x \norm y\) | Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(\le\) | \(\ds 1\) | Since $x, y \in \OO$ |
Hence:
- $x y \in \OO$
$\Box$
By Subring Test it follows that $\OO$ is a subring of $R$.
Since $1_R \in S$ and $1_R$ is the unity of $R$ then $1_R$ is the unity of $\OO$.
By Division Ring has No Proper Zero Divisors then $R$ has no proper zero divisors.
Hence $\OO$ has no proper zero divisors.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.4$ Algebra: Proposition $2.4.1$