  Using the multi-fluid approach, we investigate streaming and thermal
instabilities of the electron-ion plasma with homogeneous cold cosmic rays
drifting perpendicular to the background magnetic field. Perturbations across
the magnetic field are considered. The back-reaction of cosmic rays resulting
in the streaming instability is taken into account. The thermal instability is
shown not to be subject to the action of cosmic rays in the model under
consideration. The dispersion relation for the thermal instability has been
derived which includes sound velocities of plasma and cosmic rays, Alfv\'{e}n
and cosmic ray drift velocities. The relation between these parameters
determines the kind of thermal instability from Parker's to Field's type
instability. The results obtained can be useful for a more detailed the
investigation of electron-ion astrophysical objects such as galaxy clusters
including the dynamics of streaming cosmic rays.
