An inertial frame of reference is where F=ma applies. So if there are no outside forces there is no change in momentum, so a=0.
A non-inertial reference frame is an accelerated reference frame. If I place a ball on an accelerated trolley, it’ll appear to be rolling towards the back, from the reference frame fixed to the trolley. So F=ma isn’t true. We can modify it account for the acceleration by introducing a fictitious force. If the trolley is accelerating with a0 if F=0 the motion is a0 for the ball in the opposite direction so F-a0m=ma. If F=0 we have a=-a0. So the acceleration of the ball is the same magnitude and in the opposite direction of the acceleration of the trolley.
A rotating frame of reference is also like that but we will have different fictitious forces: centrifugal, Coriolis and Euler.
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