Quotient Theorem for Monomorphisms

Theorem

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 $\psi$.

Proof

By definition, $\struct {K, \circ_R}$ and $\struct {L, \circ_S}$ are inverse completions of $\struct {R, \circ_R}$ and $\struct {S, \circ_S}$ respectively.

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So by the Extension Theorem for Homomorphisms, there is one and only one monomorphism $\psi: \struct {K, \circ_R} \to \struct {L, \circ_S$ extending $\phi$.

Thus:

$\forall x \in R, y \in R^*: \map \psi {\dfrac x y} = \dfrac {\map \phi x} {\map \phi y}$

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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.10$