The investigation of the first papers of the present series (Griv, Planer. Space Sci. 44, 579-589, 1996; Griv and Yuan, Planet. Space Sci. 44, 1185-1190, 1996) is extended to include the effect of spatial inhomogeneity, i.e. the surface density gradient of Saturn's disk of mutual-gravitating identical particles. The model of an infinitely thin inhomogeneous disk with rare physical (inelastic) collisions between particles is considered. Similar to the investigation of the first papers, the well-elaborated mathematical methods of plasma linear kinetic theory are utilized. The influence of disk inhomogeneity on the oscillation spectrum of different small-amplitude oscillations in the Saturnian ring-system is studied. Estimates are made for the frequencies and the growth rate of the oscillations. It is shown that, as a rule, in Saturn's A ring the spatial inhomogeneity reinforces the growth rate of the oscillating instability (or the resonant self-excitation of density waves), and weakens the growth rate both in the C ring and in the inner portions of the B ring. In addition, the investigation of waves and their instabilities in the plane of Saturn's rings is extended by taking into account the density wave excitation/absorption at Lindblad's resonances and the weak effect of the finite but not large ring disk thickness. (C) 1997 Elsevier Science Ltd.