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Plasmon resonances and the plasmon-induced field enhancement (FE) in sodium nanoring dimers are investigated by time-dependent density functional theory. For larger separations, the optical absorption, the induced charge response and the frequency dependent current demonstrate that there are two capacitive coupling plasmon modes. One feature of FE is that, in the surface region of the nanoring, it has a very large maximum. Another feature of FE is that, along the perpendicular bisector of the line segment joining the two nanoring center points in the middle region of the nanoring dimers, it has maxima. With the decrease of the gap distance, because of the electrons tunneling across the dimer junction and screening, collective excitation modes are changed, and the charge transfer plasmon modes emerge in the nanoring dimers. FE induced by any plasmon modes decreases in the gap region. Moreover, corresponding to different gap distances, the high-energy plasmon resonance peak almost does not shift, because this plasmon mode is mainly the collective excitation as a result of interactions among degenerate individual electronic states. [ABSTRACT FROM AUTHOR]