Stability properties of the free‐streaming mode (space‐charge wave) in a relativistic annular electron beam with radius R0 propagating through a dielectric loaded waveguide is investigated, in connection with the Cherenkov radiation. The stability analysis is carried out within the framework of the linearized Vlasov‐Maxwell equations for an electron distribution function, in which all electrons have a Lorentzian distribution in the axial canonical momentum. One of the most important features of the analysis is that, for some ranges of physical parameters, a strong mode coupling between the vacuum dielectric waveguide and free streaming modes occurs, exhibiting possibilities of a Cherenkov radiation. The typical maximum growth rate of instability is a few percent of c/R0, where c is the speed of light in vacuo. However, the growth rate and bandwidth of instability are substantially reduced by increasing the axial momentum spread.