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In sodium nanoring dimers, plasmon resonances and the plasmon-induced field enhancement are investigated by time-dependent density functional theory. The optical absorption, the induced charge response, and the frequency dependent current demonstrate that the main plasmon resonance modes are the charge transfer plasmon mode and the bonding dimer plasmon mode (BDP). Moreover, there are also two small hybridized plasmon modes. The induced field enhancement of each spatial region depends on the gap distance and the plasmon mode. For the narrow gap, the field enhancement at different positions of the straight line segments between two nanorings is almost uniformly distributed. However, for large separations, along the axial direction, the field enhancement gradually decreases in the region within the radius of the sodium atom. Then, the change of the field enhancement is nonlinear. For different plasmon modes, there is a different number of field enhancement extrema. The largest extreme value is located in the middle region. These findings are expected to play an important role in designing plasmonic nanostructures for practical applications that require coupled metallic nanoparticles with enhanced electric fields.