Is there a notation to designate the quotient in integral domain?

Question

Let $R$ be an integral domain and $p$ be a nonzero element of $R$.

Let $p,r \in R$ such that $p$ is nonzero and $p|r$. Then, there exists a unique $q\in R$ such that $r=pq$.

Is there a notation which designates this $q$?

Answer 1

It is often written as $q = r/p$.

We can define the field of fractions as $S^{-1}R$ where $S = R \setminus \{0\}$ and it consists of equivalence classes denoted by $\displaystyle \frac{r}{s}$, where $\displaystyle \frac{r}{s} = \frac{p}{t}$ if and only if there exists $u \in S$ such that $(rt - sp)u = 0$.

So in this case $\displaystyle \frac{q}{1} = \frac{r}{p}$ as $(q p - r) \cdot 1 = qp - r = 0$ since we had $r = qp$ to begin with.

So the notation above comes from looking at rings/fields of fractions.

The key here is to notice that $r/p$ is not the same thing as $rp^{-1}$!