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

Then:

$\dfrac a b + \dfrac c d = \dfrac {a \circ d + b \circ c} {b \circ d}$

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

## Proof

First we demonstrate the operation has the specified property:

 $\ds \frac a b + \frac c d$ $=$ $\ds a \circ b^{-1} + c \circ d^{-1}$ Definition of Division Product $\ds$ $=$ $\ds a \circ b^{-1} \circ d \circ d^{-1} + c \circ d^{-1} \circ b \circ b^{-1}$ Definition of Inverse Element and Definition of Identity Element under $\circ$ $\ds$ $=$ $\ds \paren {a \circ d} \circ \paren {d^{-1} \circ b^{-1} } + \paren {b \circ c} \circ \paren {d^{-1} \circ b^{-1} }$ Definition of Commutative Operation $\ds$ $=$ $\ds \paren {a \circ d + b \circ c} \circ \paren {b \circ d}^{-1}$ Definition of Distributive Operation $\circ$ over $+$ $\ds$ $=$ $\ds \frac {a \circ d + b \circ c} {b \circ d}$ Definition of Division Product

Notice that this works only if $\struct {R, +, \circ}$ is commutative.

$\blacksquare$