# Embedding Division Ring into Quotient Ring of Cauchy Sequences

## Contents

## Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\mathcal C$ be the ring of Cauchy sequences over $R$

Let $\mathcal N = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0}$

Let $\norm {\, \cdot \,}:\mathcal C \, \big / \mathcal N \to \R_{\ge 0}$ be the norm on the quotient ring $\mathcal C \, \big / \mathcal N$ defined by:

- $\displaystyle \forall \sequence {x_n} + \mathcal N: \norm {\sequence {x_n} + \mathcal N} = \lim_{n \mathop \to \infty} \norm {x_n}$

Let $\phi: R \to \mathcal C \, \big / \mathcal N$ be the mapping from $R$ to the quotient ring $\mathcal C \, \big / \mathcal N$ defined by:

- $\forall a \in R: \map \phi a = \sequence {a, a, a, \dotsc} + \mathcal N$

where $\sequence {a, a, a, \dotsc} + \mathcal N$ is the left coset in $\mathcal C \, \big / \mathcal N$ that contains the constant sequence $\sequence {a, a, a, \dotsc}$.

Then:

- $\phi$ is a distance-preserving ring monomorphism.

## Proof

By the definition of a distance-preserving mapping and a ring monomorphism it has to be shown that:

- $(1): \quad \phi$ is a homomorphism.
- $(2): \quad \phi$ is an injection.
- $(3): \quad \phi$ is distance-preserving.

### $(1): \quad \phi$ is a homomorphism

By definition, $\phi$ is the composition of two mappings:

- $\phi = q \circ \phi'$

where:

- $\text{(a)}: \quad \phi': R \to \mathcal C$, defined by: $\forall a \in R, \map {\phi'} a = \sequence {a, a, a, \dotsc}$
- $\text{(b)}: \quad q$ is the quotient mapping $q: \mathcal C \to \mathcal C \, \big / \mathcal N$ defined by: $\map q {\sequence {x_n} } = \sequence {x_n} + \mathcal N$

By Embedding Normed Division Ring into Ring of Cauchy Sequences, $\phi'$ is a ring monomorphism.

By Quotient Ring Epimorphism is Epimorphism, then $q$ is a ring epimorphism.

By Composition of Ring Homomorphisms is Ring Homomorphism then the composition $\phi = q \circ \phi'$ is a ring homomorphism

$\Box$

### $(2): \quad \phi$ is an injection

Let $a, b \in R$.

Suppose $\map \phi a = \map \phi b$.

Then:

\(\displaystyle \sequence {a, a, a, \dotsc} + \mathcal N\) | \(=\) | \(\displaystyle \sequence {b, b, b, \dotsc} + \mathcal N\) | Definition of $\phi$ | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sequence {a, a, a, \dotsc} - \sequence {b, b, b, \dotsc}\) | \(\in\) | \(\displaystyle \mathcal N\) | Left Cosets are Equal iff Product with Inverse in Subgroup | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sequence {a - b, a - b, a - b, \dotsc}\) | \(\in\) | \(\displaystyle \mathcal N\) | Ring operations on Ring of Cauchy Sequences | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} {a - b}\) | \(=\) | \(\displaystyle 0\) | Definition of $\mathcal N$ |

By Constant Sequence Converges to Constant in Normed Division Ring then:

- $\displaystyle \lim_{n \mathop \to \infty} {a - b} = a - b$

Hence $a-b = 0$.

The result follows.

$\Box$

### $(3): \quad \phi$ is distance-preserving

Let $a, b \in R$.

Then:

\(\displaystyle \norm {\phi \paren {a} - \phi \paren {b} }\) | \(=\) | \(\displaystyle \norm {\phi \paren {a - b} }\) | $\phi$ is a homomorphism | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \norm {\sequence {a - b, a - b, a - b, \dots} + \mathcal N}\) | Definition of $\phi$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{n \mathop \to \infty } \norm {a - b}\) | Definition of $\norm {\, \cdot \,}$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \norm {a - b}\) | Constant Sequence Converges to Constant in Normed Division Ring |

$\blacksquare$

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 3.2$: Completions, Problem $85$