  We analyzed the effects of a toroidal magnetic field in the formation of
several magnetized accretion tori, dubbed as ringed accretion disks (RADs),
orbiting around one central Kerr supermassive Black Hole (SMBH) in AGNs, where
both corotating and counterotating disks are considered. Constraints on tori
formation and emergence of RADs instabilities, accretion onto the central
attractor and tori collision emergence, are investigated. The results of this
analysis show that the role of the central BH spin-mass ratio, the magnetic
field and the relative fluid rotation and tori rotation with respect the
central BH, are crucial elements in determining the accretion tori features,
providing ultimately evidence of a strict correlation between SMBH spin, fluid
rotation and magnetic fields in RADs formation and evolution. More specifically
we proved that magnetic field and disks rotation are in fact strongly
constrained, as tori formation and evolution in RADs depend on the toroidal
magnetic fields parameters. Eventually this analysis identifies specific
classes of tori, for restrict ranges of magnetic field parameter, that can be
observed around some specific SMBHs identified by their dimensionless spin.
