Applying the superposition principle, is the most simple method for finding the closed-loop gain of the inverter circuit. As long as the opamp is operated within its linear region we can set the resulting voltage Vn between ground and the inverting terminal Vn=0 (in reality it is as small as some µVolts).
Superposition:
(1) Vout=0: Vn1=Vi*Rf/(Ri+Rf)
(2) Vi=0: Vn2=Vout*Ri/(Rf+Ri)
(3) Setting Vn=Vn1+Vn2=0 we arrive at Vout/Vi=-Rf/Ri