Field of Quotients is Divisible Module

Theorem
Let $A$ be an integral domain.

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

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

Proof
Let $a \in A$ be a non zero divisor, and $x, y \in A$ such that $y \ne 0$.

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

By definition of integral domain:
 * $a \ne 0$

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

It follows that:
 * $a \cdot \dfrac x {a y} = \dfrac x y$

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