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 an 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.

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, $\forall x, y \in R, z, w \in R^*$:

Thus $\psi: K \to L$ is a monomorphism.