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

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

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

Proof
$$ $$ $$ $$ $$

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