# Inverse of Division Product

## Theorem

Let $\struct {R, +, \circ}$ be a commutative ring with unity.

Let $\struct {U_R, \circ}$ be the group of units of $\struct {R, +, \circ}$.

Let $a, b \in U_R$.

Then:

$\paren {\dfrac a b}^{-1} = \dfrac {1_R} {\paren {a / b}} = \dfrac b a$

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

## Proof

 $\displaystyle \frac {1_R} {\paren {a / b} }$ $=$ $\displaystyle 1_R / \paren {a \circ b^{-1} }$ Definition of Division Product $\displaystyle$ $=$ $\displaystyle 1_R \circ \paren {a \circ b^{-1} }^{-1}$ Definition of Division Product $\displaystyle$ $=$ $\displaystyle \paren {a \circ b^{-1} }^{-1}$ Definition of Identity Element of $\circ$ $\displaystyle$ $=$ $\displaystyle \paren {\frac a b}^{-1}$ Definition of Division Product $\displaystyle$ $=$ $\displaystyle \paren {a \circ b^{-1} }^{-1}$ Definition of Division Product $\displaystyle$ $=$ $\displaystyle \paren {b^{-1} }^{-1} \circ a^{-1}$ Inverse of Group Product $\displaystyle$ $=$ $\displaystyle b \circ a^{-1}$ Inverse of Group Inverse $\displaystyle$ $=$ $\displaystyle \frac b a$ Definition of Division Product

$\blacksquare$