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 $x,y \in A$ with $y \neq 0$.

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

By definition of integral domain $a \neq 0$.

Thus $\frac{x}{ay}$ is defined in $\map{\operatorname{Quot}}{A}$.

It follows, that $a \cdot \frac{x}{ay} = \frac{x}{y}$.

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