Division notation in ring Let $a,b\in R$ where $R$ is a ring.
Is there any problem/ambiguity with this notation $\frac{ab}{a}=b$, where $a\neq 0$?
What are the minimal conditions (e.g. commutative, UFD, etc) that we need such that the above makes sense?
Thanks for any help.

In domains (for instance polynomial rings over domains) this is used quite frequently: one often finds an expression such as
$$
\frac{X^n-1}{X-1}
$$
to denote $X^{n-1}+X^{n-2}+\dots+X+1$ or, more generally,
$$
\frac{f(X)}{X-a}
$$
where $f(X)$ is a polynomial and $a$ is a root thereof, so this denotes a well defined polynomial.
This makes sense in the field of quotients, so it's just a slight abuse of language. With this convention, the equality
$$
\frac{ab}{a}
$$
always holds in a domain, provided $a\ne0$. Other examples are found when dealing with a greatest common divisor $d$ of $a$ and $b$: writing
$$
\frac{a}{d}
$$
in such cases is quite frequently found. Again, this stands for the (unique) element $c$ such that $cd=a$ and we can perform the operation in the field of quotients.
When the ring is not a domain, in particular if it is not commutative, such a notation is to be carefully avoided: we might have $ab=0$ with $b\ne0$, so $ab/a=b$ would even be false. Even if $a$ is invertible, the notation might be interpreted both as $a^{-1}ab=b$, but also as $aba^{-1}$ which may well be different from $b$.
When talking about rings of quotients, say $S^{-1}R$, where $R$ is a commutative ring and $S$ is a multiplicative set, the notation $a/s$ ($a\in R$ and $s\in S$) has a special meaning and $ab/a=b$ should be avoided as well, even if $a\in S$, unless the canonical map $R\to S^{-1}R$ is injective. This is the case when $R$ is a domain, but then we are essentially back to the situation at the beginning.
