Observations suggest the existence of both hot and cold matter in the centers of active galactic nuclei. Recent spectral models require a major fraction of power to be dissipated in the hot matter. We study the case when the hot matter forms a corona around a standard cold alpha-disk. In particular, we investigate the case when a major fraction, f, of the power released when the cold matter accretes is transported to and dissipated in the corona. This has major effects on the cold disk, making it colder, more geometrically thin, denser, and having larger optical depths. One important consequence is the disappearance of the effectively optically thin zone as well as of the radiation pressure dominated zone for values off sufficiently closed to unity. The disappearance of the radiation pressure dominated zone will result in a cold disk with only a gas pressure dominated zone that is stable against thermal and viscous instabilities. We also show that the pressure (and the radiation) from the corona will only affect the surface layers of the cold disk. Our results disagree with those of other recent work on accretion disks with coronae. We find those works to be based on unphysical assumptions.