How to define the notation of irreducible fraction?

Question

If we write:
$a_1, a_2 \in Z$,
$a = \frac{a_2}{a_1}$,
then by this statement $a$ can be integer or fraction.
I would like to write that after choosing vlues for $a$ and $b$ we have to simplify it to the statement:
$a = \frac{a_2}{a_1} = \frac{\alpha_2}{\alpha_1}$, where $\frac{\alpha_2}{\alpha_1}$ is written in the simplest way so it have to be irreducible fraction when $a \notin Z$.