The concentration dependence of the conformations of ring polymers is investigated by lattice Monte Carlo simulations and compared with that of linear polymers. The relative radii of gyration of linear polymers follow a universal master curve as a function of the scaled concentration for various chain lengths, with a scaling relationship ⟨R_g²⟩ ∼ ϕ^–0.25, which is consistent with scaling theory and neutron scattering experiments. Ring polymers of different lengths also follow a universal behavior with a broad crossover to a scaling behavior ⟨R_g²⟩ ∼ ϕ^–0.59 for long chains. The scaling relationship between the concentration dependence and the chain-length dependence of the radius of gyration implies ⟨R_g²⟩ ∼ N^0.72, indicating highly collapsed conformations for long-chain ring polymers in the melt.