We consider ringed accretion disks (RADs), representing models of aggregates of corotating and counterrotating toroids orbiting a central Kerr super-massive blackhole (SMBH). We comment on the system of two-tori governed by the polytropic equation of state and including a toroidal magnetic field. We found the RADslead-ing function describing the RADinner structure and governing the distribution of orbiting toroidal structures and the emergence of the (hydro-mechanical) instabilities in the disk. We perform this analysis first in pure hydrodynamical models by considering one-specie perfect fluid toroids and then by considering the contribution of the toroidal magnetic field.