Quotient Theorem for Monomorphisms
Theorem
Let $K, L$ be quotient fields of integral domains $\left({R, +_R, \circ_R}\right), \left({S, +_S, \circ_S}\right)$ respectively.
Let $\phi: R \to S$ be a monomorphism.
Then there is one and only one monomorphism $\psi: K \to L$ extending $\phi$, and:
- $\displaystyle \forall x \in R, y \in R^*: \psi \left({\frac x y}\right) = \frac {\phi \left({x}\right)} {\phi \left({y}\right)}$
Also, if $\phi$ is a ring isomorphism, then so is $\psi$.
Proof
By definition, $\left({K, \circ_R}\right)$ and $\left({L, \circ_S}\right)$ are inverse completions of $\left({R, \circ_R}\right)$ and $\left({S, \circ_S}\right)$ respectively.
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So by the Extension Theorem for Homomorphisms, there is one and only one monomorphism $\psi: \left({K, \circ_R}\right) \to \left({L, \circ_S}\right)$ extending $\phi$.
Thus:
- $\displaystyle \forall x \in R, y \in R^*: \psi \left({\frac x y}\right) = \frac {\phi \left({x}\right)} {\phi \left({y}\right)}$
By the Extension Theorem for Isomorphisms, $\psi$ is an isomorphism if $\phi$ is.
Thus, $\forall x, y \in R, z, w \in R^*$:
| \(\displaystyle \psi \left({\frac x z +_R \frac y w}\right)\) | \(=\) | \(\displaystyle \psi \left({\frac {\left({x \circ_R w}\right) +_R \left({y \circ_R z}\right)} {z \circ_R w} }\right)\) | Addition of Division Products | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\phi \left({\left({x \circ_R w}\right) +_R \left({y \circ_R z}\right)}\right)} {\phi \left({z \circ_R w}\right)}\) | Definition of $\psi$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\left({\phi \left({x}\right) \circ_S \phi \left({w}\right)}\right) +_S \left({\phi \left({y}\right) \circ_S \phi \left({z}\right)}\right)} {\phi \left({z}\right) \circ_S \phi \left({w}\right)}\) | Morphism Property | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\phi \left({x}\right)} {\phi \left({z}\right)} +_S \frac {\phi \left({y}\right)} {\phi \left({w}\right)}\) | Addition of Division Products | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \psi \left({\frac x z}\right) +_S \psi \left({\frac y w}\right)\) | Definition of $\psi$ |
Thus $\psi: K \to L$ is a monomorphism.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 23$: Theorem $23.10$
