# Equality of Division Products

## 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 $a, b \in R, c, d \in U_R$.

Then:

$\dfrac a c = \dfrac b d \iff a \circ d = b \circ c$

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

## Proof

 $\displaystyle \frac a c$ $=$ $\displaystyle \frac b d$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle a \circ c^{-1}$ $=$ $\displaystyle b \circ d^{-1}$ Definition of Division Product $\displaystyle \leadstoandfrom \ \$ $\displaystyle a \circ c^{-1} \circ c \circ d$ $=$ $\displaystyle b \circ d^{-1} \circ c \circ d$ Definition of Cancellable Element of $U_R$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \paren {a \circ d} \circ \paren {c^{-1} \circ c}$ $=$ $\displaystyle \paren {b \circ c} \circ \paren {d^{-1} \circ d}$ Definition of Commutative Operation $\displaystyle \leadstoandfrom \ \$ $\displaystyle a \circ d$ $=$ $\displaystyle b \circ c$ Definition of Identity Element and Definition of Inverse Element

$\blacksquare$

Alternatively, a proof can be built using Addition of Division Products.