Quotient Theorem for Monomorphisms

From ProofWiki
Jump to navigation Jump to search

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.


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

\(\displaystyle \map \psi {\frac x z +_R \frac y w}\) \(=\) \(\displaystyle \map \psi {\frac {\paren {x \circ_R w} +_R \paren {y \circ_R z} } {z \circ_R w} }\) Addition of Division Products
\(\displaystyle \) \(=\) \(\displaystyle \frac {\map \phi {\paren {x \circ_R w} +_R \paren {y \circ_R z} } } {\map \phi {z \circ_R w} }\) Definition of $\psi$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {\map \phi x \circ_S \map \phi w} +_S \paren {\map \phi y \circ_S \map \phi z} } {\map \phi z \circ_S \map \phi w}\) Morphism Property
\(\displaystyle \) \(=\) \(\displaystyle \frac {\map \phi x} {\map \phi z} +_S \frac {\map \phi y} {\map \phi w}\) Addition of Division Products
\(\displaystyle \) \(=\) \(\displaystyle \map \psi {\frac x z} +_S \map \psi {\frac y w}\) Definition of $\psi$

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

$\blacksquare$


Sources