  The stability of dissipative Taylor-Couette flows with an axial stable
density stratification and a prescribed azimuthal magnetic field is considered.
Global nonaxisymmetric solutions of the linearized MHD equations with toroidal
magnetic field, axial density stratification and differential rotation are
found for both insulating and conducting cylinder walls. Flat rotation laws
such as the quasi-Kepler law are unstable against the nonaxisymmetric
stratorotational instability (SRI). The influence of a current-free toroidal
magnetic field depends on the magnetic Prandtl number Pm: SRI is supported by
Pm > 1 and it is suppressed by Pm \lsim 1. For too flat rotation laws a smooth
transition exists to the instability which the toroidal magnetic field produces
in combination with the differential rotation. This nonaxisymmetric azimuthal
magnetorotational instability (AMRI) has been computed under the presence of an
axial density gradient. If the magnetic field between the cylinders is not
current-free then also the Tayler instability occurs and the transition from
the hydrodynamic SRI to the magnetic Tayler instability proves to be rather
complex. Most spectacular is the `ballooning' of the stability domain by the
density stratification: already a rather small rotation stabilizes magnetic
fields against the Tayler instability. An azimuthal component of the resulting
electromotive force only exists for density-stratified flows. The related
alpha-effect for magnetic SRI of Kepler rotation appears to be positive for
negative d\rho/dz <0.
