Quotient_Theorem_for_Monomorphisms

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 $\psi$.