The stability of two rigidly rotating magnetic fluids separated by a cylindrical interface and stressed by a timely oscillating axial magnetic field is investigated. Only axisymmetric disturbances are considered. The interface admits both mass and heat transfer. Weak viscous effects on the interface are taken into account so that their contributions are demonstrated in the boundary conditions. The solution of the boundary value problem leads to a transcendental differential equation. It includes a periodic coefficient together with modified Bessel functions of operators involved as their arguments. In the absence of rotation and under the assumption of small amplitude of the harmonic magnetic field, the characteristic equation is analyzed by means of Whittaker's perturbation technique to determine the transition curves which separate stable from unstable solutions. While in the presence of rotation, the method of multiple-time scales is adopted to investigate the necessary and sufficient conditions for stability. The analysis results in the resonance cases as well as the nonresonance cases. In order to simplify the analysis, the periodic solutions are only considered. Therefore, stability is discussed through the marginal state. Furthermore, the rotation is considered as small. The analytical results are numerically confirmed.