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

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