Field of Quotients is Divisible Module

Theorem

Let $D$ be an integral domain.

Let $\map {\operatorname {Quot} } D$ be the field of quotients of $D$.

Then $\map {\operatorname {Quot} } D$ is a divisible $D$-module.

Proof

Let $a \in D$ be a non zero divisor.

Let $x, y \in D$ such that $y \ne 0$.

Then $\dfrac x y \in \map {\operatorname {Quot} } D$.

By definition of integral domain:

$a \ne 0$

Thus $\dfrac x {a y}$ is defined in $\map {\operatorname {Quot} } D$.

It follows that:

$a \cdot \dfrac x {a y} = \dfrac x y$

Thus $\map {\operatorname {Quot} } D$ is a divisible $D$-module.

$\blacksquare$