The stability of an electrically conductive and incompressible fluid between coaxial rotating cylinders in the presence of a radial density gradient, as well as a circular magnetic field, is studied. The effects of gravity and of the heat generated through viscous and magnetic dissipation is neglected. An eigenvalue problem is formulated for rotationally symmetrical disturbances. Detailed calculations are made for the case of small spacing between the two cylinders. Three dimensionless parameters, each of which contains only one of the three (momentum, thermal, and magnetic) diffusivities, are defined. Parametric equations for surfaces of marginal stability are given, and several planes cutting such surfaces are drawn for the discussions of the roles played by the diffusivities. It is found that both the momentum diffusivity and the magnetic diffusivity are capable of playing the dual role (i.e., the role of stabilization and destabilization simultaneously) in the problem.