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.

The notation "$/a$" is a shorthand for the inverse of $a$, so one necessary feature is that $a$ is a unit in $R$, that is, there exists a $c$ such that $ac=ca=1$ and then you could write $a^{-1}=\frac{1}{a}=c$.
The second thing is that it is necessary to consider commutativity of the ring with this notation. There is no problem when the ring is commutative. But when the ring is noncommutative, $\frac{ab}{a}$ is ambiguous, because it does not tell us what order $c$ is being multiplied with $ab$. It could potentially mean $a^{-1}ab$ or it could be $aba^{-1}$. The first one reduces to just $b$ automatically, but there is no reason for $b=aba^{-1}$.
For example, in the quaternions, $iji^{-1}=-j\neq j$.
