Electromagnetic ring resonators composed of noncontacting metal nanoparticles are analyzed as building blocks of future plasmonic optical systems. Closed-form expressions for the electromagnetic modes are obtained, and conditions under which such structures exhibit low or high radiative losses are found. Most modes for subwavelength structures exhibit strongly suppressed radiative losses. Possible applications for subwavelength resonators are discussed. Nearest-neighbor tight-binding models are shown to be of severely circumscribed validity.