Definition:Division Product

Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

Let $\left({U_R, \circ}\right)$ be the group of units of $\left({R, +, \circ}\right)$.

Then we define the following notation:


 * $\forall x \in U_R, y \in R$, we have:


 * $\dfrac y x = y \circ \left({x^{-1}}\right) = \left({x^{-1}}\right) \circ y$

This is referred to as $y$ divided by $x$.

We also write (out of space considerations) $y / x$ for $\dfrac y x$.

This notation is usually used when $\left({R, +, \circ}\right)$ is a field.

Caution
We do not usually use this notation for a ring (with unity) which is not commutative, as it would not be straightforward to determine whether $\dfrac y x$ means $y \circ \left({x^{-1}}\right)$ or $\left({x^{-1}}\right) \circ y$.