sound proofing ceiling
There are two types of noise you might be likely to want to mitigate.
1) Noise from talking or playing music or something like that.
2) Noise from thumping on the floor.
Type one you might substantially reduce by lifting the carpet, putting down several layers of 30# felt, another layer of subfloor and putting the carpet back. High pitched sounds are relatively easy to absorb with mass, which is what the 30# felt is. They also make a sound absorbing material that looks like felt but works better -- don't know how much it costs. Insulation in the joists will help a little bit, but not much.
Noise type 2 you can only mitigate by having a separate ceiling joist. Sound that gets to the subfloor will go directly to the joist, through the joist to the drywall, and radiate back out.