  In this paper and a companion work, we report on the first global numerical
simulations of self-gravitating magnetized tori, subject in particular to the
influence of the magnetorotational instability (MRI). In this work, paper I, we
restrict our calculations to the study of the axisymmetric evolution of such
tori. Our goals are twofold: (1) to investigate how self-gravity influences the
global structure and evolution of the disks; and (2) to determine whether
turbulent density inhomogeneities can be enhanced by self-gravity in this
regime.
  As in non self-gravitating models, the linear growth of the MRI is followed
by a turbulent phase during which angular momentum is transported outward. As a
result, self-gravitating tori quickly develop a dual structure composed of an
inner thin Keplerian disk fed by a thicker self-gravitating disk, whose
rotation profile is close to a Mestel disk. Our results show that the effects
of self-gravity enhance density fluctuations much less than they smooth the
disk, and giving it more coherence. We discuss the expected changes that will
occur in 3D simulations, the results of which are presented in a companion
paper.
