Pulley inertia moment of inertia

The important thing is the difference in tension between the ropes on either side of the pulley.

So you will have $(T_2-T_1) R = I_{text {pulley}}alpha$ where $T_1$ and $T_2$ are the tensions on either side of the pulley, $R$ is the radius of the pulley, $I_{text {pulley}}$ the moment of inertia of the pulley and $alpha$ the angular acceleration of the pulley.

Also, if the rope rolls without slipping about a frictionless pully, α = a/R where a is the acceleration of objects attached to the ends of the rope (one object moves up and the other down).