Feedback loop-gain(beta) derivation for the inverting amplifier?
Definition: The feedback factor beta is defined as the portion of the output voltage that is fed back to the differential opamp input (input source grounded). Therefore, regarding beta there is no difference between the non-inverting and inverting configuration. In both cases, we have 

$$\beta = \frac{R_{in}}{R_{in} + R_f}$$

Using the general formula for the closed-loop voltage gain \$A_{cl}\$ of an amplifier with feedback

$$A_{cl} = k \cdot \frac{A_{ol}}{1 + A_{ol} \cdot \beta}$$ 

with open-loop gain \$A_{ol}\$ and the input damping 

$$ k = -\frac{R_f}{R_{in} + R_f}$$

we get (for \$A_{ol} \to\infty\$) 

$$Acl = -\frac{k}{\beta} = -\frac{R_f}{R_{in}}$$

EDIT: Comment/explanation: The "input damping" factor \$k\$ is defined as "portion of the input voltage arriving at the diff. opamp input" (negative because \$k \cdot V_{in}\$ arrives at the inv. terminal).