This paper describes a theoretical and computational study of the threshold and saturation conditions of the electromagnetic proton cyclotron ring instability. The initial proton velocity distribution is modeled as a Maxwellian in the direction parallel to the ambient magnetic field and as a relatively cold ring in the plane perpendicular to the field. Linear Vlasov theory and one-dimensional hybrid simulations are used to examine and confirm previously derived scaling laws for the maximum instability growth rate and maximum value of the fluctuating magnetic field energy density. Linear theory yields a new instability threshold condition relating the dimensionless ring perpendicular speed and the dimensionless ring parallel temperature. Simulation results show that cessation of fluctuation growth is due to the heating of the ring protons in the parallel direction and to the reduction of the perpendicular kinetic energy of the ring; at saturation these parameters typically lie near values predicted by a linear threshold condition. The threshold condition is predicted to provide an upper bound for proton ring kinetic energies observed in space plasmas.¿ 1997 American Geophysical Union