  Gravitational instability (GI) features in several aspects of protostellar
disk evolution, most notably in angular momentum transport, fragmentation, and
the outbursts exemplified by FU Ori and EX Lupi systems. The outer regions of
protostellar discs may also be coupled to magnetic fields, which could then
modify the development of GI. To understand the basic elements of their
interaction, we perform local 2D ideal and resistive MHD simulations with an
imposed toroidal field. In the regime of moderate plasma beta, we find that the
system supports a hot gravito-turbulent state, characterised by considerable
magnetic energy and stress and a surprisingly large Toomre parameter $Q~10$.
This result has potential implications for disk structure, vertical thickness,
ionisation, etc. Our simulations also reveal the existence of long-lived and
dense `magnetic islands' or plasmoids. Lastly, we find that the presence of a
magnetic field has little impact on the fragmentation criterion of the disk.
Though our focus is on protostellar disks, some of our results may be relevant
for the outer radii of AGN.
