Quotient Theorem for Monomorphisms
Theorem
Let $K, L$ be fields of quotients of integral domains $\struct {R, +_R, \circ_R}, \struct {S, +_S, \circ_S}$ respectively.
Let $\phi: R \to S$ be a monomorphism.
Then there is one and only one monomorphism $\psi: K \to L$ extending $\phi$, and:
- $\forall x \in R, y \in R^*: \map \psi {\dfrac x y} = \dfrac {\map \phi x} {\map \phi y}$
Also, if $\phi$ is a ring isomorphism, then so is $\psi$.
Proof
By definition, $\struct {K, \circ_R}$ and $\struct {L, \circ_S}$ are inverse completions of $\struct {R, \circ_R}$ and $\struct {S, \circ_S}$ respectively.
this is not the definition
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So by the Extension Theorem for Homomorphisms, there is one and only one monomorphism $\psi: \struct {K, \circ_R} \to \struct {L, \circ_S$ extending $\phi$.
Thus:
- $\forall x \in R, y \in R^*: \map \psi {\dfrac x y} = \dfrac {\map \phi x} {\map \phi y}$
By the Extension Theorem for Isomorphisms, $\psi$ is an isomorphism if $\phi$ is.
Thus, $\forall x, y \in R, z, w \in R^*$:
| \(\ds \map \psi {\frac x z +_R \frac y w}\) | \(=\) | \(\ds \map \psi {\frac {\paren {x \circ_R w} +_R \paren {y \circ_R z} } {z \circ_R w} }\) | Addition of Division Products | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \phi {\paren {x \circ_R w} +_R \paren {y \circ_R z} } } {\map \phi {z \circ_R w} }\) | Definition of $\psi$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {\map \phi x \circ_S \map \phi w} +_S \paren {\map \phi y \circ_S \map \phi z} } {\map \phi z \circ_S \map \phi w}\) | Morphism Property | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \phi x} {\map \phi z} +_S \frac {\map \phi y} {\map \phi w}\) | Addition of Division Products | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \psi {\frac x z} +_S \map \psi {\frac y w}\) | Definition of $\psi$ |
Thus $\psi: K \to L$ is a monomorphism.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.10$