Ai cores raise profits for colonies they govern.
You can have as many as you want, since they aren't limited by the Administrator perk.
They only really rebel if you try to take them off and that only happens with Alpha AI cores.
They can help diminish wants for drugs and even more immediate needs like supplies or organics by how high leveled they are.
They DO cause factions to inspect your colonies if they have them.
You CANNOT remove alpha AI. No matter where you put it, you can no longer remove them.
Overall they are just good business, IF you can bribe/fight against the people that look into your colonies.
Basically what you should do is give to other factions on their comms in exchange for money and rep, and then later on, you start a colony with that money you got from previous AI dealings!