  The instability of a quasi-Kepler flow in dissipative Taylor-Couette systems
under the presence of an homogeneous axial magnetic field is considered with
focus to the excitation of nonaxisymmetric modes and the resulting angular
momentum transport. The excitation of nonaxisymmetric modes requires higher
rotation rates than the excitation of the axisymmetric mode and this the more
the higher the azimuthal mode number m. We find that the weak-field branch in
the instability map of the nonaxisymmetric modes has always a positive slope
(in opposition to the axisymmetric modes) so that for given magnetic field the
modes with m>0 always have an upper limit of the supercritical Reynolds number.
In order to excite a nonaxisymmetric mode at 1 AU in a Kepler disk a minimum
field strength of about 1 Gauss is necessary. For weaker magnetic field the
nonaxisymmetric modes decay. The angular momentum transport of the
nonaxisymmetric modes is always positive and depends linearly on the Lundquist
number of the background field. The molecular viscosity and the basic rotation
rate do not influence the related {\alpha}-parameter. We did not find any
indication that the MRI decays for small magnetic Prandtl number as found by
use of shearing-box codes. At 1 AU in a Kepler disk and a field strength of
about 1 Gauss the {\alpha} proves to be (only) of order 0.005.
