  We explore streaming instabilities of the electron-ion plasma with
relativistic and ultra-relativistic cosmic rays in the background magnetic
field in the multi-fluid approach. Cosmic rays can be both electrons and ions.
The drift speed of cosmic rays is directed along the magnetic field. In
equilibrium, the return current of the background plasma is taken into account.
One-dimensional perturbations parallel to the magnetic field are considered.
The dispersion relations are derived for transverse and longitudinal
perturbations. It is shown that the back-reaction of magnetized cosmic rays
generates new instabilities one of which has the growth rate that can approach
the growth rate of the Bell instability. These new instabilities can be
stronger than the cyclotron resonance instability. For unmagnetized cosmic
rays, the growth rate is analogous to the Bell one. We compare two models of
the plasma return current in equilibrium with three and four charged
components. Some difference between these models is demonstrated. For
longitudinal perturbations, an instability is found in the case of
ultra-relativistic cosmic rays. The results obtained can be applied to
investigation of astrophysical objects such as the shocks by supernova
remnants, galaxy clusters, intracluster medium and so on, where interaction of
cosmic rays with turbulence of the electron-ion plasma produced by them is of a
great importance for the cosmic-ray evolution.
