Addition of Division Products

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 $$a, c \in R, b, d \in U_R$$. Then:


 * $$\frac a b + \frac c d = \frac {a \circ d + b \circ c} {b \circ d}$$

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

The operation $$+$$ is well-defined.

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

Proof

 * First we demonstrate the operation has the specified property:

$$ $$ $$ $$ $$

Notice that this works only if $$\left({R, +, \circ}\right)$$ is commutative.


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

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

Then:

$$ $$ $$

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

Hence:

$$ $$ $$

Thus:

$$ $$ $$ $$

showing that $$+$$ is indeed well-defined.