  In this paper, we consider regularity criteria of a class of 3D axially
symmetric MHD-Boussinesq systems without magnetic resistivity or thermal
diffusivity. Under some Prodi-Serrin type critical assumptions on the
horizontal angular component of the velocity, we will prove that strong
solutions of the axially symmetric MHD-Boussinesq system can be smoothly
extended beyond the possible blow-up time $T_\ast$ if the magnetic field
contains only the horizontal swirl component. No a priori assumption on the
magnetic field and the temperature fluctuation is imposed.
