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 $z$.

The operation $+$ is well-defined.

That is:
 * $\dfrac a b = \dfrac {a'} {b'}, \dfrac c d = \dfrac {c'} {d'} \implies \dfrac a b + \dfrac c d = \dfrac {a'} {b'} + \dfrac {c'} {d'}$

Proof
First we demonstrate the operation has the specified property:

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

Now we show that $+$ is well-defined.

Let $a, c, a', c' \in D, b, d, b', d' \in D^*$ such that $\dfrac a b = \dfrac {a'} {b'}$ and $\dfrac c d = \dfrac {c'} {d'}$.

Then:

Similarly, $c \circ d' = c' \circ d$.

Hence:

Thus:

showing that $+$ is indeed well-defined.