Inverse of Division Product

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

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

Let $a, b \in U_R$.

Then:
 * $\displaystyle \left({\frac a b}\right)^{-1} = \frac {1_R} {\left({a / b}\right)} = \frac b a$

where $\dfrac x z$ is defined as $x \circ \left({z^{-1}}\right)$, that is, $x$ divided by $z$.