Embedding Division Ring into Quotient Ring of Cauchy Sequences

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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:

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


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:

$\quad \quad \quad \forall a \in R: \phi \paren {a} = (a,a,a,\dots) + \mathcal {N}$

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

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 of $\phi$, it is the composition of two mappings:

$\phi = q \circ \phi'$

where:

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

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

By Quotient Mapping 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 $\phi \paren {a} = \phi \paren {b}$:

\(\displaystyle \implies \ \ \) \(\displaystyle \tuple {a,a,a,\dots} + \mathcal {N}\) \(=\) \(\displaystyle \tuple {b,b,b,\dots} + \mathcal {N}\) $\quad$ Definition of $\phi$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \tuple {a,a,a,\dots} - \tuple {b,b,b,\dots}\) \(\in\) \(\displaystyle \mathcal {N}\) $\quad$ Left Cosets are Equal iff Difference in Subgroup $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \tuple {a-b, a-b, a-b,\dots}\) \(\in\) \(\displaystyle \mathcal {N}\) $\quad$ Ring operations on ring of Cauchy sequences $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \lim_{n \to \infty} {a-b}\) \(=\) \(\displaystyle 0\) $\quad$ Definition of $\mathcal {N}$ $\quad$

By Constant Rule for Convergent Sequences then:

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

Hence $a-b = 0$.

The result follows.

$\Box$


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

Let $a, b \in R$.

Then:

\(\displaystyle \norm {\phi \paren {a} - \phi \paren {b} }\) \(=\) \(\displaystyle \norm {\phi \paren {a - b} }\) $\quad$ $\phi$ is a homomorphism $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \norm {\tuple {a-b, a-b, a-b, \dots} + \mathcal {N} }\) $\quad$ Definition of $\phi$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \to \infty } \norm {a-b }\) $\quad$ Definition of $\norm {\,\cdot\,}$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \norm {a-b }\) $\quad$ Constant Rule for Convergent Sequences $\quad$

$\blacksquare$


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