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We analysed the effects of a toroidal magnetic field in the formation of several magnetized accretion tori, dubbed as ringed accretion discs (RADs), orbiting around one central Kerr supermassive black hole (SMBH) in active galactic nuclei (AGNs), where both corotating and counterotating discs are considered. Constraints on tori formation and emergence of RADs instabilities, accretion on to 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 discs 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.