  This paper presents numerical linear stability analysis of a cylindrical
Taylor-Couette flow of liquid metal carrying axial electric current in a
generally helical external magnetic field. Axially symmetric disturbances are
considered in the inductionless approximation corresponding to zero magnetic
Prandtl number. Axial symmetry allows us to reveal an entirely new
electromagnetic instability. First, we show that the electric current passing
through the liquid can extend the range of helical magnetorotational
instability (HMRI) indefinitely by transforming it into a purely
electromagnetic instability. Two different electromagnetic instability
mechanisms are identified. The first is an internal pinch-type instability,
which is due to the interaction of the electric current with its own magnetic
field. Axisymmetric mode of this instability requires a free-space component of
the azimuthal magnetic field. When the azimuthal component of the magnetic
field is purely rotational and the axial component is nonzero, a new kind of
electromagnetic instability emerges. The latter driven by the interaction of
electric current with a weak collinear magnetic field in a quiescent fluid
gives rise to a steady meridional circulation coupled with azimuthal rotation.
