Is there a mathematical definition for the "divisibility" of rational numbers?

Question

The term divisibility usually refers to integer numbers only.

I want to define the divisibility of a rational number $q$ by an integer number $z$ as follows:

$q$ is divisible by $z$ if and only if $m$ is divisible by $z$, where $\frac{m}{n}$ is the simplest form of $q$.

For example: $\frac{91}{10}$ is divisible by $7$.

Is there a mathematical notation or terminology that defines this type of "divisibility"?

---

This issue has occurred to me while reading a question about the divisibility of $4^n+10\cdot9^{2n-2}$ by $7$.

It is obviously not true for $n=0$, unless we can generalize the definition of divisibility for rational numbers as stated above.

Thanks

Answer 1

You can look the notion of discrete valuation, if p is a prime, write $x=p^ia/b$ gcd(a,b)=1, $v_p(x)=i$. Here you can say that $x$ is divisible by p if $v_p(x)>0$, if $x=m/n, gcd(m,n)=1$ write $m=p^ia$, gcd(a,p)=1 $x$ is divisible by p in your sense if and only if $v_p(x)>0$

Answer 2

Given a rational number $m/n$ one can go to the localization $R$ (which is a PID) of $\mathbf{Z}$ at a suitable prime $p$ not a factor of $n$, and so $m/n$ will be an element of $R$. Being a PID one can define divisibility there.