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In this paper we present the results of numerical calculations of electromagnetic properties for cylindrical ring resonators (RRs) with rectangular cross-section and a dielectric permittivity corresponding to silicon ε = 12. The calculation of the scattering spectrum (Radar Cross Section) and the field distribution of the modes were performed at in-plane polarized light excitation. The presence of four side walls in the RRs determines a richer spectrum of eigenmodes in comparison with cylindrical whispering gallery modes of disk resonators and creates more possibilities and diversity for their practical applications.