Concentric charged rings:

Pictured to the right are three charged rings - we can set the net charges of each of the rings and watch how the charges distribute themselves. Since the rings are conductors, the charges want to move away from each other as much as possible - and there should be NO electric field within the rings.

First, set the charges and see what induced charges occur on the inner and outer surfaces of the rings ... and below, we can discuss your observations.

Observations/conclusions:

Red ring: Notice that all of the charge moves to the outside of the ring (regardless of the sign of the charge). If you draw an imaginary "gaussian ring" with a radius less than the red ring .. there should be no Electric field at the surface, so there can't be any charge enclosed.

Green ring: Notice that charge on the inside surface of the green ring has the same magnitude as the red charge, but is opposite in sign. Thus, a gaussian ring within the green ring should have no Electric field, and thus no net enclosed charge - thus the inner green charge must cancel with the outer red charge! What about the charge on the outside of the green ring? Well, notice that if you add the outside green to the inside green, you get the net charge of the green ring. The charges of the green room can move around to avoid each other, but they must be conserved!

Blue ring: The arguments for the blue ring will be the same as the green ring ... inside blue charge will be equal and opposite to the outside green charge. Outside blue charge plus inside blue charge will be the net blue charge of the ring. Also notice that the outside charge is the sum of all three ring charges!