The theory of rotating ion exospheres in dipolar magnetic field geometries is used to predict some effect of the rings on the inner plasmaspheres of the ringed planets Jupiter, Saturn, and Uranus. Plasma exospheres consist of four types of particles: 1) ballistic particles with trajectories that intersect the exobase in the same hemisphere twice, 2) trapped particles with trajectories that do not intersect the exobase in either hemisphere 3) escaping particles which travel from the exobase to the conjugate exobase and 4) incoming particles which similarly arrive at the exobase from the conjugate exobase. The analysis carried out herein assumes a purely ionospheric source. It is found that if rings affect the trapped plasma only, the cold plasma density at saturn will be either reduced or increased by a factor of ~2 at L = 1.65, depending on whether the ring populates or empties the trapped trajectories. The ionospheric plasma density at Uranus similarly depends on the properties of Uranus' rings. At Jupiter, however, ionospheric particles in the inner plasmasphere are primarily on ballistic trajectories which do not intersect the rings except possibly at longitudes where the rings are far from the magnetic equator (due to the non-dipolar magnetic components of Jupiter's field). When the properties of the rings as sources and sinks of ions are more accurately determined, these conclusions may have to be altered. However, it is found that if all cold plasma particles on trapped orbits are absorbed and the rings are not a source of plasma, this simple exospheric theory can explain the ionospheric density profile inferred from the Pioneer 11 Saturn radio occultation experiment, in which a localized peak in the profile was observed on field lines threading the Guerin division between the C and D rings. The results of the imminent Voyager radio occultation observations will provide more data for quantitative analyses with the exospheric model.