  The magnetohydrodynamic stability of axially unbounded cylindrical flows is
considered which contain a toroidal magnetic background field with the same
radial profile as the linear azimuthal velocity. Chandrasekhar (1956) has shown
for ideal fluids the stability of this configuration if the Alfven velocity of
the field equals the velocity of the background flow. It is demonstrated for
magnetized Taylor-Couette flows at the Rayleigh line, however, that for finite
diffusivity such flows become unstable against nonaxisymmetric perturbations
where the critical magnetic Reynolds number of the rotation rate does not
depend on the magnetic Prandtl number Pm if Pm much << 1. In order to study
this new diffusive azimuthal magnetorotational instability, flows and fields
with the same radial profile but with different amplitudes are considered. For
Pm << 1 the instability domain with the weakest fields and the slowest rotation
rates lies below the Chandrasekhar line of equal amplitudes for Alfven velocity
and rotation velocity. We find that then the lines of marginal instability
scale with the Reynolds number and the Hartmann number. The minimum values of
the field strength and the rotation rate which are needed for the instability
(slightly) grow for more and more flat rotation. Finally, the corresponding
electric current of the background field becomes so strong that the Tayler
instability (which even exists without rotation) also appears in the
bifurcation map at small Hartmann numbers displacing after all the azimuthal
magnetorotational instability.
