I'm looking for a definition like this：
In integral domains, if $ b\neq 0 $ and $b \mid a $, we define $\frac{a}{b}$ (or $a/b$ ) to be the only element $x$ such that $bx=a$.
I see many people on MSE use this division in their proofs (see 1 and 2) without declaration, but I can't find any information about it in formal writings such as algebra books from Artin, Lang, or Hungerford.
My question is that: Is the definition suitable?
If this division is well-defined, what useful properties does it have?
Any direct answers or links to other places are much appreciated.
For any ring $R$ and $a\in R$, the map $\mu_a:R\to R,\: x\mapsto ax$ is a morphism from the additive group $(R,+)$ to itself (indeed, it is even a morphism of (right) $R$-modules $R_R\to R_R$). An integral domain can be characterised as a commutative ring with identity element such that $\mu_a:R\to R$ is injective for all $a\ne0$. For $b\in \mathrm{im}(\mu_a)$, it certainly does make sense to define $b/a := \mu_a^{-1}(b)$.
Most of the usual properties can be proved by working with the maps $\mu_a$. For example, since $\mu_{ab}(x) = (ab)x = a(bx) = \mu_a(\mu_b(x))$ for all $x$, we have $\mu_{ab} = \mu_a\circ\mu_b$, and hence $(c/b)/a = \mu_a^{-1}(\mu_b^{-1}(c)) = (\mu_{b}\mu_a)^{-1}(c) = \mu_{ba}^{-1}(c) = c/(ba)\;$ (whenever defined).