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:


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

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

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

Note
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 $$\frac y x$$ means $$y \circ \left({x^{-1}}\right)$$ or $$\left({x^{-1}}\right) \circ y$$.