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:

$$\forall x \in R, y \in R^*: \psi \left({\frac x y}\right) = \frac {\phi \left({x}\right)} {\phi \left({y}\right)}$$

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:

$$\forall x \in R, y \in R^*: \psi \left({\frac x y}\right) = \frac {\phi \left({x}\right)} {\phi \left({y}\right)}$$

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.