Well, The title i guess is enough to get what i'm looking for:
I'm wondering if we can define equality of let's say $a$ and $b$ that the devision of $a$ over $b$ or $b$ over $a$ is $1$ : $$a=b \implies \frac{a}{b}=\frac{b}{a}=1$$ if that is true then why it doesn't holds for $0$ case : $$\frac{0}{0}\neq1$$
$$\frac 00 \neq 1.$$ Indeed $\frac 00$ is undefined. But $a = b$ when $a = b=0$.
So you cannot say $a = b \implies \frac ab = 1$.
What you can say is $$\frac ab = 1 \implies a = b$$
It's the "if and only if* that's problematic, for the reason at the top.