  We consider the effect of toroidal magnetic fields on hydrodynamically stable
Taylor-Couette differential rotation flows. For current-free magnetic fields a
nonaxisymmetric m=1 magnetorotational instability arises when the magnetic
Reynolds number exceeds O (100). We then consider how this `azimuthal
magnetorotational instability' (AMRI) is modified if the magnetic field is not
current-free, but also has an associated electric current throughout the fluid.
This gives rise to current-driven Tayler instabilities (TI) that exist even
without any differential rotation at all. The interaction of the AMRI and the
TI is then considered when both electric currents and differential rotation are
present simultaneously. The magnetic Prandtl number Pm turns out to be crucial
in this case. Large Pm have a destabilizing influence, and lead to a smooth
transition between the AMRI and the TI. In contrast, small Pm have a
stabilizing influence, with a broad stable zone separating the AMRI and the TI.
In this region the differential rotation is acting to stabilize the Tayler
instabilities, with possible astrophysical applications (Ap stars). The growth
rates of both the AMRI and the TI are largely independent of Pm, with the TI
acting on the timescale of a single rotation period, and the AMRI slightly
slower, but still on the basic rotational timescale. The azimuthal drift
timescale is ~20 rotations, and may thus be a (flip-flop) timescale of stellar
activity between the rotation period and the diffusion time.
