  We study $r$-modes of a rotating magnetized neutron star, assuming a magnetic
dipole field whose axis is aligned with the axis of rotation. We approach the
problem by applying a singular perturbation theory to the oscillations of
rotating stars. In this treatment, we divide the star into a thin surface
magnetic layer and a non-magnetic core. We integrate linearized ideal MHD
equations in the surface magnetic layer and non-magnetic oscillation equations
in the core, and match the two integrations at the interface to obtain a
complete solution. For a polytropic neutron star model of mass $M=1.4M_\odot$
and radius $R=10^6$cm, the magnetic dipole field becomes effective on the modal
properties of the $r$-modes only when the field strength $B_S$ is much greater
than $10^{14}$G. We also find that the damping effects caused by very short
magnetic perturbations in the surface layer are not important for the $r$-mode
instability of rapidly rotating neutron stars if the field strength $B_S$ is
smaller than $10^{12}$G.
