Linear, steady, axisymmetric flow of an electrically conducting homogeneous fluid confined within a rigid, rotating electrically insulated cylinder is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The applied magnetic field and the rotation vector are aligned normal to the horizontal boundaries. The magnetic Reynolds number, the Rossby number, and the Ekman number are assumed to be much less than unity. The dynamics of the inviscid interior and vertical boundary layers are investigated as functions of the rotational magnetic interaction parameter α2(=σB2/2ρΩ) which measures the ratio between the magnetic force and the Coriolis force. If α2 ≪1, the flow behaves as a nonmagnetic rotational flow to dominant order with Stewartson’s E1/3 and E1/4 double layer structures and the electric currents pumped by the Ekman–Hartmann layer return through the interior. At the other extreme, if α2≫E−1/3 the flow behaves as a strongly magnetic nonrotational flow with a single vertical layer (≈E1/4 α−1/2) of parabolic structure. The intermediate range 1≪α2≪E−1/3, which is a transition from weakly magnetic to the strongly magnetic flows, is characterized by a quadruple vertical boundary layer structure: (1) a nonmagnetic E1/3 layer, (2) a nonmagnetic E1/4α1/2 layer, (3) a magnetic α−2 layer to transport the electric current vertically, and (4) a magnetic E1/4α−1/2 layer with axial scale E1/4α3/2 to feed to feed the current from the Ekman–Hartmann layers into the α−2 layer.