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.
this is not the definition
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
To discuss this proof in more detail, feel free to use the talk page.
If you are able to fix this page, then when you have done so you can remove this instance of {{Questionable}} from the code.
If you would welcome a second opinion as to whether your correction is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
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, $\forall x, y \in R, z, w \in R^*$:
| \(\displaystyle \psi \left({\frac x z +_R \frac y w}\right)\) | \(=\) | \(\displaystyle \psi \left({\frac {\left({x \circ_R w}\right) +_R \left({y \circ_R z}\right)} {z \circ_R w} }\right)\) | $\quad$ Addition of Division Products | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\phi \left({\left({x \circ_R w}\right) +_R \left({y \circ_R z}\right)}\right)} {\phi \left({z \circ_R w}\right)}\) | $\quad$ Definition of $\psi$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\left({\phi \left({x}\right) \circ_S \phi \left({w}\right)}\right) +_S \left({\phi \left({y}\right) \circ_S \phi \left({z}\right)}\right)} {\phi \left({z}\right) \circ_S \phi \left({w}\right)}\) | $\quad$ Morphism Property | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\phi \left({x}\right)} {\phi \left({z}\right)} +_S \frac {\phi \left({y}\right)} {\phi \left({w}\right)}\) | $\quad$ Addition of Division Products | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \psi \left({\frac x z}\right) +_S \psi \left({\frac y w}\right)\) | $\quad$ Definition of $\psi$ | $\quad$ |
Thus $\psi: K \to L$ is a monomorphism.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 23$: Theorem $23.10$
