  We investigate stationary, self-gravitating, magnetised disks (or tori)
around black holes. The models are obtained by numerically solving the coupled
system of the Einstein equations and the equations of ideal
general-relativistic magnetohydrodynamics. The mathematical formulation and
numerical aspects of our approach are similar to those reported in previous
works modeling stationary self-gravitating perfect-fluid tori, but the
inclusion of magnetic fields represents a new ingredient. Following previous
studies of purely hydrodynamical configurations, we construct our models
assuming Keplerian rotation in the disks and both spinning and spinless black
holes. We focus on the case of a toroidal distribution of the magnetic field
and build a large set of models corresponding to a wide range of values of the
magnetisation parameter, starting with weakly magnetised disks and ending at
configurations in which the magnetic pressure dominates over the thermal one.
In all our models, the magnetic field affects the equilibrium structure of the
torus mainly due to the magnetic pressure. In particular, an increasing
contribution of the magnetic field shifts the location of the maximum of the
rest-mass density towards inner regions of the disk. The total mass of the
system and the angular momentum are affected by the magnetic field in a complex
way, that depends on the black hole spin and the location of the inner radius
of the disk. The non-linear dynamical stability of the solutions presented in
this paper will be reported elsewhere.
